Thermal Properties of Matter

1. Temperature

Temperature is the physical quantity that determines the degree of hotness or coldness of a body. It decides the direction of heat flow when two bodies are placed in contact.

According to the kinetic theory of matter, temperature is proportional to the average kinetic energy of molecules of the substance.

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2. Heat

Heat is a form of energy that is transferred between two systems or between a system and its surroundings due to a difference in temperature.

It is important to note that heat is energy in transit. Once the energy is absorbed by a body, it becomes part of the body's internal energy.

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3. Units of Heat and Temperature

• SI unit of heat energy → Joule (J)
• SI unit of temperature → Kelvin (K)
• Common temperature scale → Degree Celsius (°C)
• Another scale → Fahrenheit (°F)

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4. Heat Flow Illustration

Heat flows spontaneously from a body at higher temperature to a body at lower temperature.

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5. Key Differences Between Heat and Temperature

1. Concept of Temperature Measurement

Measurement of temperature is the process of assigning a numerical value to the degree of hotness or coldness of a body. This is done by bringing the body into thermal equilibrium with a thermometer and relating the temperature to a measurable physical property that changes uniformly with temperature.

This principle is based on the Zeroth Law of Thermodynamics which states:

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2. Thermometric Property

Temperature cannot be measured directly. Instead, it is inferred from a thermometric property — a physical property that changes systematically with temperature.

Common thermometric properties include:

• Expansion of liquids (mercury or alcohol thermometer)
• Gas pressure at constant volume
• Electrical resistance of a conductor
• Thermoelectric emf in thermocouples
• Volume change of gases

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3. Illustration of a Thermometer

A thermometer measures temperature by observing the change in a thermometric property.

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4. Temperature Scales

Temperature scales are defined using two fixed reference points:

• Ice Point: Temperature of melting ice
• Steam Point: Temperature of boiling water at standard atmospheric pressure

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5. Relationship Between Fahrenheit and Celsius Scales

Simplified form:

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6. Absolute Temperature Scale (Kelvin)

The Kelvin scale is the absolute temperature scale used in SI units. It is based on the behavior of an ideal gas and is independent of the properties of any specific substance.

The lowest possible temperature is absolute zero (0 K), at which molecular motion theoretically ceases.

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Important Aspects

• Temperature is not measured directly; it is inferred from a thermometric property.
• Accurate measurement requires thermal equilibrium between the thermometer and the body.
• Different temperature scales use different calibration intervals between fixed reference points.
• The gas thermometer forms the basis of the absolute Kelvin temperature scale.
• The Kelvin scale is the standard temperature scale used in physics and thermodynamics.

Statement of Boyle’s Law

Boyle’s Law describes the relationship between the pressure and volume of a gas when its temperature and mass remain constant. It states that the pressure of a gas is inversely proportional to the volume occupied by the gas under constant temperature conditions.

This means that if the volume of a gas decreases, its pressure increases, and if the volume increases, the pressure decreases. The law therefore explains how gases respond when they are compressed or allowed to expand.

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Mathematical Expression

When the temperature and the amount of gas remain constant:

or

If a gas changes from one state to another while obeying Boyle’s law, then

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Molecular Interpretation

According to the kinetic theory of gases, pressure is produced by the continuous collisions of gas molecules with the walls of the container. When the volume of a gas is reduced while keeping the temperature constant, the molecules have less space to move.

As a result, collisions with the container walls occur more frequently, which increases the pressure of the gas. Conversely, when the volume increases, the frequency of collisions decreases and the pressure falls.

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Graphical Representation

The pressure–volume graph of a gas at constant temperature is a rectangular hyperbola.

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Important Points

• Boyle’s law represents an isothermal process, meaning the temperature remains constant.
• It applies most accurately to gases at low pressure and high temperature, where gases behave nearly like ideal gases.
• When the volume of a gas decreases, the rate of molecular collisions with the container walls increases, causing the pressure to rise.
• A graph of P versus V gives a rectangular hyperbola.
• A graph of P versus 1/V is a straight line passing through the origin.

Statement of Charles’ Law

Charles’ Law describes how the volume of a gas changes with temperature when the pressure and the amount of gas remain constant. It states that the volume of a fixed mass of gas is directly proportional to its absolute temperature.

In simple terms, if the temperature of a gas increases, the gas expands and occupies a larger volume. If the temperature decreases, the gas contracts and its volume becomes smaller, provided the pressure remains unchanged.

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Mathematical Form

When pressure and mass of the gas remain constant:

or

If the gas changes from one state to another while following Charles’ law, the relation becomes:

Temperature must always be expressed in Kelvin (K).

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Molecular Interpretation

According to the kinetic theory of gases, temperature is a measure of the average kinetic energy of gas molecules. When the temperature of a gas increases, the molecules move faster and collide more energetically with the walls of the container.

If pressure is kept constant, the gas must expand so that the rate of collisions remains balanced. This expansion results in an increase in the volume of the gas.

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Graphical Representation

The graph between volume and absolute temperature is a straight line passing through the origin.

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Concept of Absolute Zero

When the volume–temperature graph is extended backward, it predicts that the volume of an ideal gas would become zero at a certain temperature. This temperature is called absolute zero.

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Important Points

• Charles’ law represents an isobaric process, meaning the pressure remains constant.
• Temperature must always be expressed on the Kelvin scale.
• Increase in temperature raises the average kinetic energy of gas molecules.
• To maintain constant pressure, the gas expands and occupies a larger volume.
• The V–T graph is a straight line through the origin when temperature is measured in Kelvin.
• Charles’ law, together with Boyle’s law, forms the foundation of the Ideal Gas Equation.

Ideal Gas and Its Equation

An ideal gas is a theoretical gas that follows all gas laws perfectly under every condition of temperature and pressure. In such a gas, the molecules are assumed to have negligible volume and there are no intermolecular forces acting between them.

The relationship connecting the macroscopic properties of a gas—pressure, volume, and temperature—is known as the ideal-gas equation. It combines the results of Boyle’s law, Charles’ law, and other gas laws into a single mathematical expression.

where:

• P = Pressure of the gas
• V = Volume occupied by the gas
• n = Number of moles of the gas
• R = Universal gas constant
• T = Absolute temperature

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Meaning of Absolute Temperature

Absolute temperature is the temperature measured on the Kelvin scale. This scale begins at the lowest possible temperature known as absolute zero.

On the Kelvin scale, temperature is directly proportional to the average kinetic energy of gas molecules. Therefore, the higher the temperature, the greater the molecular motion within the gas.

When the temperature approaches 0 Kelvin, the thermal motion of molecules becomes extremely small. This theoretical lowest temperature is called absolute zero.

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Graphical Representation

For a fixed amount of gas, the product \(PV\) is directly proportional to the absolute temperature.

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Physical Significance

The ideal-gas equation shows that the macroscopic properties of a gas are closely related to the microscopic motion of its molecules. As temperature increases, the average kinetic energy of molecules increases, leading to changes in pressure or volume.

This equation therefore forms a bridge between thermodynamics and kinetic theory of gases.

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Important Points

• The ideal-gas equation combines Boyle’s law and Charles’ law into a single relation.
• Temperature must always be measured in the Kelvin scale.
• The equation works accurately for gases at high temperature and low pressure.
• Absolute temperature provides a natural thermodynamic scale used in physics.
• The equation connects measurable quantities of a gas with the microscopic motion of its molecules.

Concept of Absolute Zero

Absolute zero is the lowest possible temperature on the thermodynamic temperature scale. It represents the point at which the thermal energy of a system becomes minimum.

At this temperature, the random motion of molecules approaches its smallest possible value. Although molecular motion does not completely stop due to quantum mechanical effects, the thermal kinetic energy of molecules becomes extremely small.

The Kelvin scale is designed so that its zero corresponds to this fundamental lower limit of temperature.

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Origin of the Concept

The concept of absolute zero arises from the study of Charles’ law. When the volume–temperature graph of a gas is extended backward, it predicts that the volume of an ideal gas would become zero at a certain temperature.

This temperature corresponds to approximately −273.15°C, which is defined as 0 Kelvin on the absolute temperature scale.

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Graphical Interpretation

Extrapolating the volume–temperature graph of a gas predicts zero volume at −273.15°C.

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Relation with Absolute Temperature

The concept of absolute zero led to the development of the Kelvin temperature scale, which is widely used in thermodynamics and physics.

On this scale, temperature directly reflects the average kinetic energy of molecules in a substance.

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Important Points

• Absolute zero is the starting point of the Kelvin temperature scale.
• It corresponds to −273.15°C on the Celsius scale.
• The concept arises from extrapolating the gas laws, especially Charles’ law.
• At absolute zero, the thermal kinetic energy of molecules becomes minimum.
• Absolute zero cannot be reached exactly in practice, but temperatures very close to it can be achieved in laboratories.
• The idea of absolute zero provides the physical foundation of the absolute temperature scale.

Concept of Thermal Expansion

Thermal expansion refers to the tendency of a substance to increase in its dimensions when its temperature rises. When a material is heated, the particles that make up the substance begin to vibrate more vigorously, increasing the average distance between them.

As a result of this increased separation, the overall size of the material becomes larger. Conversely, when the temperature decreases, the vibration of particles reduces and the substance contracts.

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Microscopic Explanation

At higher temperatures, atoms and molecules gain additional thermal energy and oscillate with greater amplitude about their equilibrium positions. These larger oscillations increase the average spacing between particles, which leads to an expansion of the material.

The effect of thermal expansion is observed in all states of matter, though the extent of expansion varies depending on intermolecular forces.

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Types of Thermal Expansion in Solids

In solid materials, thermal expansion can occur in different ways depending on which dimension changes.

• Linear Expansion – change in the length of a solid rod or wire when temperature changes.
  \[ \frac{\Delta L}{L} = \alpha \Delta T \]
• Areal Expansion – change in the surface area of a solid plate with temperature.
  \[ \frac{\Delta A}{A} = 2\alpha \Delta T \]
• Volumetric Expansion – change in the volume of a solid body when heated.
  \[ \frac{\Delta V}{V} = 3\alpha \Delta T \]

Here, \(\alpha\) represents the coefficient of linear expansion and \(\Delta T\) is the change in temperature.

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Illustration of Thermal Expansion

A solid rod increases in length when heated due to greater molecular vibration.

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Practical Applications

• Expansion joints are provided in bridges and railway tracks to prevent damage during temperature changes.
• Metal rims are heated before fitting onto wooden wheels so that they expand and fit tightly after cooling.
• Bimetallic strips used in thermostats rely on the difference in expansion of two metals.
• Glass containers may crack if hot liquid is poured suddenly due to unequal thermal expansion.

Definition

The coefficient of linear expansion describes how much the length of a solid changes when its temperature changes. It is defined as the fractional increase in length per unit original length for each unit rise in temperature.

In simple terms, it tells us how sensitive the length of a material is to changes in temperature. Different materials expand at different rates, which is why metals, glass, and other solids respond differently to heating.

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Mathematical Expression

where:

• \(L\) = original length of the object
• \(\Delta L\) = change in length
• \(\Delta T\) = change in temperature
• \(\alpha\) = coefficient of linear expansion

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Physical Interpretation

When a solid is heated, its atoms vibrate more vigorously about their equilibrium positions. This increased vibration slightly increases the average distance between neighboring atoms, leading to a small increase in the length of the material.

The coefficient of linear expansion quantifies this effect and allows us to predict how much a solid object will expand for a given temperature change.

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Illustration

A solid rod increases slightly in length when its temperature rises.

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Important Points

• The formula is valid only for small temperature changes.
• The material must remain within its elastic limit so that the expansion is reversible.
• Different materials have different values of the coefficient of linear expansion.
• Materials with larger values of \(\alpha\) expand more for the same temperature increase.

Definition

The coefficient of volume expansion, also known as volume expansivity, describes how the volume of a substance changes with temperature. It is defined as the fractional increase in volume per unit original volume for each unit rise in temperature, provided the temperature change is small and uniform throughout the material.

In simple terms, it tells us how much a material expands in three dimensions when its temperature increases.

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Mathematical Expression

where:

• \(V\) = original volume of the substance
• \(\Delta V\) = change in volume
• \(\Delta T\) = change in temperature
• \(\alpha_v\) = coefficient of volume expansion

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Microscopic Explanation

When a substance is heated, the molecules gain kinetic energy and vibrate more vigorously. This increased vibration causes the average separation between particles to increase.

Because the expansion occurs in all three spatial directions, the overall volume of the material increases.

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Illustration

When a solid body is heated, expansion occurs in all three dimensions, resulting in an increase in volume.

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Relation with Linear Expansion

For isotropic solids (materials that expand equally in all directions), the coefficient of volume expansion is related to the coefficient of linear expansion.

where \( \alpha \) is the coefficient of linear expansion.

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Important Points

• Volume expansion occurs in all three dimensions.
• The formula is valid only for small temperature changes.
• Different materials have different coefficients of volume expansion.
• In gases and liquids, volume expansion is generally much larger than in solids.

Concept

The coefficients of linear expansion and volume expansion are closely related for a solid body that expands equally in all directions. Such materials are called isotropic solids.

To understand the relation, consider a cube of side \(l\). When the temperature of the cube increases by \(\Delta T\), each edge of the cube expands slightly.

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Linear Expansion

According to the definition of the coefficient of linear expansion,

where \( \Delta l \) represents the small increase in the length of each side of the cube.

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Change in Volume of the Cube

Initially, the volume of the cube is

After heating, each side becomes \( l + \Delta l \), so the new volume is

Expanding this expression,

Since \( \Delta l \) is very small compared with \(l\), the terms containing \( (\Delta l)^2 \) and \( (\Delta l)^3 \) are negligible. Therefore,

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Deriving the Relation

Dividing both sides by the original volume \(V = l^3\),

Substituting \( \frac{\Delta l}{l} = \alpha_l \Delta T \),

But from the definition of volume expansion,

Comparing the two expressions,

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Illustration

Expansion occurs in all three dimensions, leading to a larger volume.

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Important Result

• For isotropic solids, the coefficient of volume expansion is three times the coefficient of linear expansion.
• This relation is valid only for small temperature changes.
• The derivation assumes that the expansion is uniform in all directions.

Concept of Thermal Stress

Thermal stress is the internal stress that develops in a material when its natural expansion or contraction due to temperature change is restricted by external constraints.

Normally, when a solid is heated it tends to expand, and when it is cooled it contracts. If the body is free to change its dimensions, this expansion or contraction occurs without any stress.

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How Thermal Stress Develops

• A solid tends to expand on heating and contract on cooling.
• If its ends are fixed or its motion is restricted, expansion cannot occur.
• This restriction produces internal forces inside the material.
• These internal forces give rise to thermal stress.

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Mathematical Expression

For a rod whose ends are rigidly fixed, the thermal stress produced due to a temperature change is given by:

where:

• \(Y\) = Young’s modulus of the material
• \(\alpha\) = coefficient of linear expansion
• \(\Delta T\) = change in temperature

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Illustration

A heated rod fixed between rigid supports develops thermal stress because expansion is prevented.

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Physical Interpretation

When the temperature of a constrained rod increases, the atoms attempt to move farther apart. Since the ends are fixed, the material experiences compressive stress internally. Similarly, if the rod is cooled while its ends are fixed, tensile stress develops.

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Important Points

• Thermal stress appears only when expansion or contraction is prevented.
• If a body is free to expand or contract, no thermal stress develops.
• Thermal stress is directly proportional to Young’s modulus, the coefficient of linear expansion, and the temperature change.
• Engineers must consider thermal stress when designing structures such as bridges and railway tracks.
• Sudden heating or cooling can cause materials like glass to crack due to thermal stress.

Definition

The specific heat capacity of a substance is the amount of heat required to raise the temperature of unit mass of that substance by one degree without causing any change in its physical state.

Different materials respond differently to heating. Some substances require a large amount of heat to produce a small rise in temperature, while others warm up quickly with relatively little heat. This behavior is described by their specific heat capacity.

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Mathematical Expression

or equivalently

where:

• \(Q\) = heat supplied to the substance
• \(m\) = mass of the substance
• \(\Delta T\) = change in temperature
• \(C\) = specific heat capacity

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Molar Specific Heat Capacity

Sometimes the quantity of substance is expressed in terms of number of moles instead of mass. In that case we define the molar specific heat capacity.

where:

• \(\mu\) = number of moles of the substance
• \(C_m\) = molar specific heat capacity

Its SI unit is J mol⁻¹ K⁻¹.

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Microscopic Explanation

When heat is supplied to a substance, the molecules gain kinetic energy and move more vigorously. This increase in molecular motion raises the temperature of the substance.

Substances with higher specific heat capacity require more energy to increase the kinetic energy of their molecules, which is why they warm up more slowly.

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Illustration

Heat supplied to a substance increases the kinetic energy of its molecules, causing the temperature to rise.

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Important Points

• Specific heat capacity is a characteristic property of a material.
• Materials with higher specific heat capacity require more heat for the same temperature rise.
• Water has a very high specific heat capacity, which helps regulate Earth's climate and biological systems.
• The formula \(Q = mC\Delta T\) is valid only when no phase change occurs.
• During melting or boiling, heat supplied changes the state of matter rather than the temperature.

Definition

The molar specific heat capacity at constant pressure, denoted by \(C_p\), is the amount of heat required to raise the temperature of one mole of a substance by one kelvin while the pressure of the system remains constant.

When heating occurs at constant pressure, the supplied heat not only increases the internal energy of the gas but also performs work in expanding the gas against the external pressure.

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Mathematical Expression

For a finite temperature change:

where:

• \(Q\) = heat supplied to the system
• \(n\) = number of moles of the substance
• \(\Delta T\) = change in temperature
• \(C_p\) = molar specific heat capacity at constant pressure

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Physical Significance

During heating at constant pressure, a gas tends to expand as its temperature rises. This expansion requires additional energy to do work against the surrounding pressure.

Because some of the supplied heat is used to perform expansion work, the heat required at constant pressure is greater than that required when the volume is kept constant.

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Illustration

When heat is supplied at constant pressure, the gas expands and does work on the surroundings.

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Important Points

• \(C_p\) represents the heat required to raise the temperature of one mole of a substance at constant pressure.
• Some of the supplied heat is used to perform expansion work.
• For gases, \(C_p\) is always greater than the molar specific heat capacity at constant volume \(C_v\).
• The relation between them for an ideal gas is \(C_p - C_v = R\).

Definition

The molar specific heat capacity at constant volume, represented by \(C_v\), is the amount of heat required to raise the temperature of one mole of a substance by one kelvin when the volume of the system remains constant.

When heating occurs at constant volume, the gas cannot expand. Therefore, the supplied heat is used entirely to increase the internal energy of the gas.

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Mathematical Expression

For a finite temperature change:

where:

• \(Q\) = heat supplied to the gas
• \(n\) = number of moles of the gas
• \(\Delta T\) = change in temperature
• \(C_v\) = molar specific heat capacity at constant volume

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Physical Significance

At constant volume, the gas cannot perform expansion work because the volume does not change. Hence, all the heat supplied goes into increasing the kinetic energy of the molecules, which increases the internal energy of the gas.

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Illustration

At constant volume, heat supplied increases the internal energy without causing expansion.

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Relation with \(C_p\)

For an ideal gas, the molar specific heat capacity at constant pressure is always greater than that at constant volume.

The difference between the two is given by the important thermodynamic relation:

where \(R\) is the universal gas constant.

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Important Points

• \(C_v\) measures the heat needed to increase temperature when volume is fixed.
• At constant volume, the gas performs no external work.
• All supplied heat increases the internal energy of the gas.
• For ideal gases, \(C_p\) is always greater than \(C_v\).
• The relation \(C_p - C_v = R\) is a fundamental result of kinetic theory.

Concept of Calorimetry

Calorimetry is the branch of thermal physics that deals with the measurement of heat exchanged during physical or chemical processes. It is based on the fundamental principle of conservation of energy.

In calorimetry experiments, the heat lost by a hotter object is measured by observing the heat gained by a colder object when both are brought into thermal contact inside a well-insulated container known as a calorimeter.

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Principle of Calorimetry

When bodies at different temperatures are placed in contact within an isolated system, heat flows from the hotter body to the colder body until both reach the same temperature, called thermal equilibrium.

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Mathematical Expression

\[ \sum Q = 0 \]

This means that the algebraic sum of heat exchanges in the system is zero. In practical calculations, it is usually written as:

If no heat escapes to the surroundings, the heat lost by the hot body exactly equals the heat gained by the colder body and the calorimeter.

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Calorimeter

A calorimeter is an insulated container used to measure heat exchange between substances. It is designed to minimize heat loss to the surroundings so that accurate measurements can be obtained.

A typical calorimeter consists of a metal container, a lid, a thermometer, and a stirrer to ensure uniform temperature throughout the mixture.

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Illustration

A calorimeter measures heat exchange by observing temperature changes in an insulated system.

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Important Points

• Calorimetry is based on the law of conservation of energy.
• Heat always flows from a higher temperature body to a lower temperature body.
• The calorimeter should be well insulated to prevent heat exchange with the surroundings.
• The final temperature reached is called the equilibrium temperature.
• The principle of calorimetry is widely used in determining specific heat capacity and latent heat.

Definition

A calorimeter is a well-insulated device used to measure the amount of heat exchanged during a thermal process. It enables accurate determination of heat transfer when bodies at different temperatures interact.

In calorimetry experiments, substances are placed inside the calorimeter so that heat exchange occurs only between the bodies within the container, while heat loss to the surroundings is minimized.

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Construction of a Simple Calorimeter

A typical laboratory calorimeter consists of several important components that help ensure accurate heat measurements.

• A thin metallic container (usually made of copper or aluminium).
• A well-fitting lid to reduce heat loss.
• A thermometer to measure temperature changes.
• A stirrer to maintain uniform temperature inside the liquid.

The metallic container is made thin so that it can quickly reach the same temperature as the substance inside.

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Working Principle

A calorimeter operates on the principle of calorimetry, which states that in an isolated system the heat lost by the hotter body equals the heat gained by the colder body until thermal equilibrium is reached.

In practical experiments, the calorimeter itself may also absorb some heat, so this heat must be included in the calculations.

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Water Equivalent

The heat absorbed by the calorimeter is often expressed in terms of its water equivalent. The water equivalent of a calorimeter is the mass of water that would absorb the same amount of heat as the calorimeter for the same temperature rise.

Using the water equivalent simplifies calorimetric calculations because the calorimeter can be treated as an equivalent mass of water.

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Illustration

A simple calorimeter consists of a metal container, thermometer, and stirrer.

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Important Points

• A calorimeter is designed to minimise heat exchange with the surroundings.
• The container is usually made of copper or aluminium because these metals conduct heat efficiently.
• Stirring ensures uniform temperature throughout the liquid.
• The calorimeter’s heat capacity must be included in calorimetric calculations.
• Calorimeters are widely used to determine specific heat capacity, latent heat, and other thermal properties of substances.

Definition

The melting point of a substance is the specific temperature at which a solid changes into a liquid at a given pressure. At this temperature, the solid and liquid phases of the substance exist together in thermal equilibrium.

During the melting process, the temperature of the substance remains constant until the entire solid has converted into liquid.

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What Happens During Melting?

When heat is supplied to a solid at its melting point, the energy does not increase the temperature immediately. Instead, the supplied heat is used to overcome the intermolecular forces holding the particles in their fixed positions within the solid structure.

Once these intermolecular forces are sufficiently weakened, the particles can move more freely and the substance enters the liquid state.

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Microscopic Explanation

In a solid, atoms or molecules occupy fixed positions in a regular structure and only vibrate about their equilibrium points. As heat is supplied, the vibrational energy of the particles increases.

At the melting point, these vibrations become strong enough to break the rigid structure of the solid, allowing the particles to move freely, which marks the transition to the liquid state.

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Illustration

A solid absorbs heat at its melting point and gradually transforms into liquid.

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Effect of Pressure

The melting point of most substances increases when the pressure applied to them increases. This happens because higher pressure favors the phase with smaller volume.

However, ice behaves differently. The melting point of ice decreases with increasing pressure because the liquid phase of water occupies a smaller volume than the solid phase.

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Important Points

• The melting point is a characteristic physical property of a substance.
• Temperature remains constant during the melting process.
• Heat supplied at the melting point is used as latent heat of fusion.
• Solid and liquid phases coexist at the melting temperature.
• Solid and liquid have the same temperature at melting point but different internal energies.

Definition

Regelation is the phenomenon in which ice melts when pressure is applied and refreezes when the pressure is removed, provided the temperature is close to or slightly below its normal melting point.

This process occurs because the melting point of ice decreases when pressure increases. As a result, applying pressure can cause a thin layer of ice to melt even when the temperature is below \(0^\circ C\).

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How Regelation Occurs

The phenomenon can be understood through the following sequence:

• When pressure is applied to ice, its melting point decreases.
• If the surrounding temperature is near the melting point, the ice melts locally.
• When the pressure is removed, the melting point returns to its normal value.
• The water formed during melting freezes again.

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Microscopic Explanation

Ice has an open crystalline structure in which molecules are arranged with relatively larger spacing compared with liquid water. When pressure is applied, this structure collapses slightly, making it easier for the molecules to move into the more compact liquid arrangement.

Consequently, the solid phase melts under pressure and refreezes once the pressure is removed.

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Illustration

A wire can pass through a block of ice due to melting under pressure and refreezing behind it.

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Applications and Examples

• A thin wire can pass through a block of ice without splitting it.
• The slow movement of glaciers over rocks occurs partly due to regelation.
• Ice skating becomes easier because pressure under the skate blade causes temporary melting.

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Important Points

• Regelation occurs because the melting point of ice decreases with increasing pressure.
• It is possible only for substances whose solid phase is less dense than the liquid phase.
• Ice is one of the most common substances that exhibit this behavior.
• The phenomenon explains several natural and practical processes involving ice.

Definition

Vaporisation is the process in which a substance changes from the liquid state to the gaseous state by absorbing heat energy. This transformation occurs when the molecules of the liquid gain sufficient energy to overcome the intermolecular forces holding them together.

Vaporisation can take place at different conditions of temperature and pressure, depending on the nature of the liquid and its surroundings.

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Modes of Vaporisation

A liquid can change into vapour in two distinct ways:

• Evaporation
  A slow process that occurs at the surface of a liquid at all temperatures. Only the molecules with sufficient kinetic energy escape from the surface.
• Boiling
  A rapid process that occurs throughout the liquid at a fixed temperature known as the boiling point.

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Boiling Point

Boiling begins when the vapour pressure of the liquid becomes equal to the external atmospheric pressure. At this stage, vapour bubbles form inside the liquid and rise to the surface.

During boiling, the temperature of the liquid remains constant even though heat continues to be supplied.

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Latent Heat of Vaporisation

The heat absorbed during the conversion of liquid into vapour at constant temperature is known as the latent heat of vaporisation.

This energy is used to overcome the intermolecular forces between the liquid molecules rather than increasing the temperature.

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Microscopic Explanation

In a liquid, molecules move randomly but remain relatively close together due to intermolecular attraction. When heat is supplied, the kinetic energy of molecules increases.

Eventually some molecules acquire enough energy to escape from the liquid and enter the gaseous state, leading to vaporisation.

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Illustration

During boiling, vapour bubbles form inside the liquid and rise to the surface.

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Important Points

• Vaporisation is the transition of a liquid into vapour.
• Evaporation occurs only at the surface and can happen at any temperature.
• Boiling occurs throughout the liquid at a fixed temperature.
• Temperature remains constant during boiling.
• Vaporisation involves a large increase in volume when the liquid becomes gas.

Definition

The boiling point of a liquid is the temperature at which the vapour pressure of the liquid becomes equal to the external (atmospheric) pressure. At this temperature, the liquid begins to transform into vapour throughout its entire volume.

Once this condition is reached, bubbles of vapour form inside the liquid and rise to the surface, indicating the onset of boiling.

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Condition for Boiling

Boiling starts when the pressure exerted by vapour molecules escaping from the liquid equals the pressure applied by the surroundings.

At this stage, vapour bubbles can form freely inside the liquid without being immediately compressed by the surrounding pressure.

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What Happens During Boiling?

When a liquid reaches its boiling point, the temperature of the liquid remains constant even though heat continues to be supplied.

The supplied heat is used as the latent heat of vaporisation, which provides the energy needed to overcome intermolecular forces and convert the liquid into vapour.

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Microscopic Explanation

Inside a liquid, molecules are held together by intermolecular attraction. As heat is supplied, the kinetic energy of molecules increases.

At the boiling point, many molecules gain enough energy to escape from these attractions, allowing vapour bubbles to form throughout the liquid and rise to the surface.

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Illustration

During boiling, vapour bubbles form throughout the liquid and rise to the surface.

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Effect of Pressure

The boiling point of a liquid depends strongly on the external pressure.

• If external pressure increases, the boiling point increases.
• If external pressure decreases, the boiling point decreases.

This is why water boils at a lower temperature on high mountains where atmospheric pressure is lower.

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Boiling vs Evaporation

• Boiling: occurs throughout the liquid at a fixed temperature.
• Evaporation: occurs only at the surface and can happen at any temperature.

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Important Points

• Boiling occurs when vapour pressure equals atmospheric pressure.
• Temperature remains constant during boiling.
• Heat supplied is used as latent heat of vaporisation.
• Boiling occurs throughout the liquid, unlike evaporation.
• The boiling point of a liquid depends on external pressure.

Definition

Sublimation is the process in which a substance changes directly from the solid state to the gaseous state without passing through the liquid phase. This transformation occurs under specific conditions of temperature and pressure.

During sublimation, molecules leave the surface of the solid and enter the vapour phase when they acquire sufficient energy to overcome the intermolecular forces holding them in the solid structure.

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Latent Heat of Sublimation

The heat absorbed by a substance when it changes directly from solid to vapour at constant temperature is called the latent heat of sublimation.

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Why Sublimation Occurs

Sublimation occurs when the vapour pressure of a solid becomes equal to or greater than the surrounding atmospheric pressure at temperatures below the melting point of the substance.

Under such conditions, molecules can escape directly from the solid surface into the gaseous state without forming a liquid phase.

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Microscopic Explanation

In a solid, particles vibrate about fixed positions in a crystal lattice. When heat is supplied, the vibrational energy increases.

If some particles gain enough kinetic energy, they can break free from the lattice and move directly into the gaseous state, producing sublimation.

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Illustration

In sublimation, a solid absorbs heat and directly converts into vapour.

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Examples of Sublimation

• Camphor
• Naphthalene
• Ammonium chloride
• Dry ice (solid carbon dioxide)
• Iodine crystals

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Important Points

• Sublimation is a direct transition from solid to gas.
• Temperature remains constant during the phase change.
• Heat absorbed is called the latent heat of sublimation.
• The process involves a large increase in volume.
• Sublimation is commonly used for purification of certain substances in chemistry.

Definition

Latent heat is the amount of heat absorbed or released by a substance during a change of state without any change in temperature. This heat is required to transform the substance from one phase to another, such as from solid to liquid or from liquid to vapour.

During a phase transition, the temperature remains constant because the supplied heat does not increase the kinetic energy of the molecules. Instead, it is used to change the internal structure of the substance.

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Why Temperature Remains Constant

When heat is supplied during a phase change, the energy is used to overcome the intermolecular forces that hold the particles together. This increases the potential energy of the molecules rather than their kinetic energy.

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Types of Latent Heat

• Latent Heat of Fusion
  The heat required to convert unit mass of a solid into liquid at its melting point without changing temperature.
• Latent Heat of Vaporisation
  The heat required to convert unit mass of a liquid into vapour at its boiling point without changing temperature.

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Mathematical Relation

where:

• \(Q\) = heat absorbed or released
• \(m\) = mass of the substance
• \(L\) = specific latent heat

The SI unit of specific latent heat is J kg⁻¹.

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Illustration

Heat supplied during phase change converts matter between solid, liquid, and gaseous states without raising temperature.

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Important Points

• Latent heat is associated with phase change, not temperature change.
• Temperature remains constant during melting and boiling.
• Heat supplied increases intermolecular separation.
• Latent heat plays an important role in many natural processes such as cloud formation and climate regulation.

Definition

Conduction is the mode of heat transfer in which thermal energy flows from a region of higher temperature to a region of lower temperature through a material medium, without any large-scale movement of the substance itself.

In this process, heat is transferred through successive collisions between neighbouring atoms or molecules. In metals, free electrons also play an important role in carrying thermal energy.

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Fourier’s Law of Heat Conduction

The rate of heat flow through a conductor depends on the temperature difference between its ends, the cross-sectional area of the conductor, and the length of the path through which heat travels.

According to Fourier’s law,

where:

• \(H\) = rate of heat flow (heat current)
• \(K\) = thermal conductivity of the material
• \(A\) = cross-sectional area of the conductor
• \(L\) = length of the conductor
• \(T_C - T_D\) = temperature difference between the ends

The constant \(K\) represents the ability of a material to conduct heat.

---

Microscopic Explanation

In solids, atoms are arranged in a lattice structure. When one part of the solid is heated, the atoms in that region vibrate more vigorously. These vibrations are transferred to neighbouring atoms through collisions, allowing thermal energy to propagate through the material.

In metals, free electrons move rapidly through the structure and transfer energy efficiently, making metals excellent conductors of heat.

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Illustration

Heat flows from the hotter region of a conductor to the colder region.

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Factors Affecting Heat Conduction

• Nature of the material (thermal conductivity)
• Temperature difference between the ends
• Cross-sectional area of the conductor
• Length of the conductor

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Important Points

• Conduction is the dominant mode of heat transfer in solids.
• Heat always flows from higher temperature to lower temperature.
• Metals such as copper and aluminium are good conductors of heat.
• Materials like wood, glass, and plastic are poor conductors (thermal insulators).
• Thermal conductivity determines how efficiently a material can conduct heat.

Definition

Convection is the mode of heat transfer in which thermal energy is transported from one region to another through the actual bulk movement of a fluid such as a liquid or gas.

Unlike conduction, where energy is transferred through molecular collisions, convection involves the physical movement of the fluid carrying heat from a hotter region to a colder region.

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How Convection Occurs

Convection arises because temperature differences in a fluid cause variations in density.

• When a portion of the fluid is heated, it expands and becomes less dense.
• The lighter, warmer fluid rises upward.
• Cooler and denser fluid sinks downward.
• This continuous circulation sets up convection currents.

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Illustration of Convection Currents

Warm fluid rises while cooler fluid sinks, creating circulating convection currents.

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Types of Convection

• Natural Convection
  Occurs due to density differences caused by temperature changes in the fluid.
• Forced Convection
  Occurs when an external device such as a fan, pump, or blower forces the fluid to move.

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Applications of Convection

• Heating of water in a vessel
• Formation of land and sea breezes
• Atmospheric circulation and wind patterns
• Cooling systems in engines and electronic devices

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Important Points

• Convection occurs only in fluids (liquids and gases).
• It results from density differences caused by temperature variations.
• Heat transfer occurs through circulation of the fluid.
• Convection plays a crucial role in weather patterns and ocean currents.
• Both natural and forced convection are widely used in engineering systems.

Definition

Radiation is the mode of heat transfer in which thermal energy is emitted and transmitted in the form of electromagnetic waves from one body to another, without the need for a material medium.

Unlike conduction and convection, radiation can occur even through empty space (vacuum). This is the mechanism by which heat from the Sun reaches the Earth.

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Nature of Thermal Radiation

Thermal radiation consists mainly of infrared electromagnetic waves produced by the thermal motion of charged particles in matter.

Every object whose temperature is above absolute zero continuously emits thermal radiation.

---

Factors Affecting Heat Radiation

The rate of heat radiation depends on several factors:

• Temperature of the body
• Surface area of the emitting body
• Nature and colour of the surface

Dark and rough surfaces emit and absorb radiation more effectively than smooth and polished surfaces.

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Stefan–Boltzmann Law

The total energy radiated per unit time from a body is proportional to the fourth power of its absolute temperature.

where:

• \(H\) = rate of heat radiation
• \(A\) = surface area of the body
• \(T\) = absolute temperature
• \(\sigma\) = Stefan–Boltzmann constant

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Illustration

Heat can travel from a hot body to a colder body through electromagnetic radiation.

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Applications of Radiation

• Heat received from the Sun
• Solar heating and solar cookers
• Thermal insulation using reflective surfaces
• Infrared heaters and thermal imaging

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Important Points

• Radiation does not require any material medium.
• It can occur through vacuum as well as transparent media.
• All bodies above absolute zero emit thermal radiation.
• Dark surfaces are better emitters and absorbers of heat radiation.
• Radiation is the only mode of heat transfer possible in outer space.

Concept of a Blackbody

A blackbody is an ideal physical object that absorbs all incident electromagnetic radiation falling on it, regardless of wavelength or direction. Because it absorbs energy perfectly, it is also considered a perfect emitter of thermal radiation.

The radiation emitted by a blackbody depends only on its absolute temperature and not on the material or surface properties of the body.

The electromagnetic radiation emitted by such an object due to its temperature is known as blackbody radiation.

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Characteristics of Blackbody Radiation

The spectrum of radiation emitted by a blackbody contains a wide range of wavelengths. The intensity of radiation varies with wavelength and temperature.

As the temperature of the body increases, two important changes occur:

• The total energy emitted increases rapidly.
• The wavelength of maximum intensity shifts towards shorter wavelengths.

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Wien’s Displacement Law

Wien’s displacement law states that the wavelength at which the intensity of radiation emitted by a blackbody is maximum is inversely proportional to its absolute temperature.

where:

• \(\lambda_{max}\) = wavelength corresponding to maximum emission
• \(T\) = absolute temperature of the body

The constant in this relation is called Wien’s constant.

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Illustration

As temperature increases, the peak of the blackbody radiation curve shifts towards shorter wavelengths.

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Practical Realisation of a Blackbody

Although a perfect blackbody does not exist in nature, a good approximation can be obtained by using a small hole in a hollow cavity. Radiation entering the hole undergoes multiple reflections and is almost completely absorbed.

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Important Points

• A blackbody is a perfect absorber and emitter of radiation.
• The radiation emitted depends only on temperature.
• Increasing temperature shifts the peak wavelength to smaller values.
• Wien’s displacement law connects peak wavelength with temperature.
• Blackbody radiation played a key role in the development of quantum physics.

Statement

The Stefan–Boltzmann law states that the total radiant energy emitted per unit time from a blackbody is directly proportional to the fourth power of its absolute temperature.

This law describes how the intensity of thermal radiation emitted by a hot object increases rapidly as its temperature rises.

---

Mathematical Expression

where

• \(H\) = total radiant energy emitted per unit time
• \(A\) = surface area of the body
• \(T\) = absolute temperature
• \(\sigma\) = Stefan–Boltzmann constant

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Net Radiation from a Body

A body not only emits radiation but also absorbs radiation from its surroundings. If the surrounding temperature is \(T_s\), the net rate of energy loss becomes

This equation gives the net heat lost by radiation.

---

Illustration

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Important Points

• Radiated energy increases rapidly with temperature (\(T^4\)).
• A larger surface area emits more thermal radiation.
• Dark surfaces radiate heat more effectively than polished surfaces.
• The law applies most accurately to an ideal blackbody.

Statement

Newton’s law of cooling states that the rate at which a body loses heat is directly proportional to the difference between the temperature of the body and the temperature of its surroundings, provided the temperature difference is small.

---

Mathematical Expression

where

• \(T\) = temperature of the body
• \(T_s\) = temperature of the surroundings
• \(k\) = cooling constant

The negative sign indicates that the temperature of the body decreases with time.

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Cooling Equation

If the mass of the body is \(m\) and its specific heat capacity is \(c\), the temperature variation with time becomes

This shows that cooling follows an exponential decay.

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Graphical Interpretation

The temperature difference \((T - T_s)\) decreases exponentially with time. A plot of \( \ln (T - T_s) \) versus time is a straight line.

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Conditions for Validity

• The temperature difference must be small.
• The surrounding temperature must remain constant.
• The nature and area of the surface must remain unchanged.

---

Applications

• Cooling of hot liquids
• Temperature estimation in forensic science
• Industrial cooling systems
• Cooling of electronic devices

A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are 5.243 m and 5.231 m respectively at \(27^\circ C\). To what temperature should the ring be heated so that it just fits the rim of the wheel?

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Concept / Theory

When a metal object is heated, its dimensions increase due to thermal expansion. For small temperature changes, the increase in length (or diameter) of a solid is given by

where \( \alpha_l \) is the coefficient of linear expansion, \(l\) is the original length, and \( \Delta T \) is the change in temperature. In this problem, the iron ring must expand until its diameter becomes equal to the diameter of the wooden rim.

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Solution

Given:

• Initial temperature \(T_1 = 27^\circ C\)
• Diameter of wooden rim \(D_{rim} = 5.243 \, m\)
• Diameter of iron ring \(D_{ring} = 5.231 \, m\)
• Coefficient of linear expansion of iron \( \alpha_l = 1.2 \times 10^{-5} \, ^\circ C^{-1} \)

For the ring to fit the rim, its diameter must increase from \(5.231\, m\) to \(5.243\, m\).

Increase in diameter:

\[ \Delta l = D_{rim} - D_{ring} \] \[ \Delta l = 5.243 - 5.231 = 0.012\, m \]

Using the linear expansion relation

\[ \frac{\Delta l}{l} = \alpha_l (T_2 - T_1) \]

Solving for temperature rise:

\[ T_2 - T_1 = \frac{\Delta l}{l \alpha_l} \]

Substituting the given values:

\[ T_2 - T_1 = \frac{0.012}{5.231 \times 1.2 \times 10^{-5}} \] \[ T_2 - T_1 \approx 191^\circ C \]

Therefore,

\[ T_2 = T_1 + 191 \] \[ T_2 = 27 + 191 = 218^\circ C \]

A sphere of 0.047 kg aluminium is placed for sufficient time in a vessel containing boiling water so that the sphere reaches \(100^\circ C\). It is then quickly transferred to a 0.14 kg copper calorimeter containing 0.25 kg water at \(20^\circ C\). The temperature of the mixture rises to \(23^\circ C\). Calculate the specific heat capacity of aluminium.

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Concept / Theory

This problem is based on the principle of calorimetry, which states that in an isolated system the total heat lost by hotter bodies equals the total heat gained by colder bodies until thermal equilibrium is reached.

In this case, the hot aluminium sphere loses heat, while the water and copper calorimeter gain heat until the final temperature becomes \(23^\circ C\).

---

Solution

Given:

• Mass of aluminium sphere \(m_{Al} = 0.047 \, kg\)
• Initial temperature of aluminium \(T_{Al} = 100^\circ C\)
• Mass of copper calorimeter \(m_{Cu} = 0.14 \, kg\)
• Mass of water \(m_w = 0.25 \, kg\)
• Initial temperature of water and calorimeter \(T_w = T_{Cu} = 20^\circ C\)
• Final equilibrium temperature \(T_f = 23^\circ C\)
• Specific heat of water \(s_w = 4.18 \times 10^3 \, J\,kg^{-1}K^{-1}\)
• Specific heat of copper \(s_{Cu} = 0.386 \times 10^3 \, J\,kg^{-1}K^{-1}\)

---

Step 1: Heat Lost by Aluminium

\[ Q_{Al} = m_{Al} \, s_{Al} (T_{Al} - T_f) \] \[ Q_{Al} = 0.047 \times s_{Al} \times (100 - 23) \] \[ Q_{Al} = 0.047 \times s_{Al} \times 77 \]

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Step 2: Heat Gained by Water and Calorimeter

\[ Q_{gain} = m_w s_w (T_f - T_w) + m_{Cu} s_{Cu} (T_f - T_{Cu}) \] Since \(T_f - T_w = 23 - 20 = 3^\circ C\), \[ Q_{gain} = 3 \left[0.25 \times 4.18 \times 10^3 + 0.14 \times 0.386 \times 10^3 \right] \]

Calculate the terms:

\[ 0.25 \times 4.18 \times 10^3 = 1045 \] \[ 0.14 \times 0.386 \times 10^3 = 54.04 \] \[ Q_{gain} = 3(1045 + 54.04) \] \[ Q_{gain} = 3 \times 1099.04 \] \[ Q_{gain} = 3297.12 \, J \]

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Step 3: Apply Calorimetry Principle

\[ Q_{Al} = Q_{gain} \] \[ 0.047 \times s_{Al} \times 77 = 3297.12 \] \[ s_{Al} = \frac{3297.12}{0.047 \times 77} \] \[ s_{Al} = \frac{3297.12}{3.619} \] \[ s_{Al} \approx 911 \, J\,kg^{-1}K^{-1} \]

When \(0.15\,kg\) of ice at \(0^\circ C\) is mixed with \(0.30\,kg\) of water at \(50^\circ C\) in a container, the final temperature of the mixture becomes \(6.7^\circ C\). Calculate the latent heat of fusion of ice. \((s_{water}=4186\,J\,kg^{-1}K^{-1})\)

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Concept / Theory

This problem is based on the principle of calorimetry. When bodies at different temperatures are mixed in an isolated system, the total heat lost by the hotter body equals the total heat gained by the colder body until thermal equilibrium is reached.

Here, the hot water loses heat, while the ice gains heat in two stages:

• Heat required to melt the ice
• Heat required to raise the temperature of melted ice to the final temperature

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Solution

Given:

• Mass of ice \(m_{ice}=0.15\,kg\)
• Initial temperature of ice \(T_{ice}=0^\circ C\)
• Mass of water \(m_w=0.30\,kg\)
• Initial temperature of water \(T_w=50^\circ C\)
• Final equilibrium temperature \(T_f=6.7^\circ C\)
• Specific heat capacity of water \(s_w=4186\,J\,kg^{-1}K^{-1}\)

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Step 1: Heat Lost by Hot Water

\[ Q_{lost}=m_w s_w (T_w-T_f) \] \[ Q_{lost}=0.30 \times 4186 \times (50-6.7) \] \[ Q_{lost}=0.30 \times 4186 \times 43.3 \] \[ Q_{lost}=54376.14\,J \]

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Step 2: Heat Gained by Ice

The ice absorbs heat in two stages.

(i) Heat required to melt ice

\[ Q_{melt}=m_{ice}L_f \] \[ Q_{melt}=0.15L_f \]

(ii) Heat required to raise temperature of melted ice

\[ Q_{raise}=m_{ice}s_w(T_f-0) \] \[ Q_{raise}=0.15 \times 4186 \times 6.7 \] \[ Q_{raise}=4206.93\,J \]

Total heat gained by ice system:

\[ Q_{gain}=0.15L_f+4206.93 \]

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Step 3: Apply Calorimetry Principle

\[ Q_{lost}=Q_{gain} \] \[ 54376.14=0.15L_f+4206.93 \] \[ 0.15L_f=54376.14-4206.93 \] \[ 0.15L_f=50169.21 \] \[ L_f=\frac{50169.21}{0.15} \] \[ L_f=334461.4\,J\,kg^{-1} \] \[ L_f \approx 3.34 \times 10^5\,J\,kg^{-1} \]

Calculate the heat required to convert 3 kg of ice at \(-12^\circ C\) into steam at \(100^\circ C\) at atmospheric pressure.
Given: \(s_{ice}=2100\,J\,kg^{-1}K^{-1}\), \(s_{water}=4186\,J\,kg^{-1}K^{-1}\), \(L_f=3.35\times10^5\,J\,kg^{-1}\), \(L_v=2.256\times10^6\,J\,kg^{-1}\).

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Concept / Theory

When a substance undergoes heating with phase changes, heat is required for both temperature rise and phase transformation.

To convert ice at \(-12^\circ C\) into steam at \(100^\circ C\), four processes occur:

• Heating ice from \(-12^\circ C\) to \(0^\circ C\)
• Melting ice at \(0^\circ C\)
• Heating water from \(0^\circ C\) to \(100^\circ C\)
• Converting water at \(100^\circ C\) into steam

The total heat required equals the sum of heat needed for these four processes.

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Solution

Given:

• Mass \(m=3\,kg\)
• Initial temperature \(T_i=-12^\circ C\)
• Final state: steam at \(100^\circ C\)
• Specific heat of ice \(s_{ice}=2100\,J\,kg^{-1}K^{-1}\)
• Specific heat of water \(s_w=4186\,J\,kg^{-1}K^{-1}\)
• Latent heat of fusion \(L_f=3.35\times10^5\,J\,kg^{-1}\)
• Latent heat of vaporisation \(L_v=2.256\times10^6\,J\,kg^{-1}\)

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Step 1: Heating Ice to \(0^\circ C\)

\[ Q_1 = m s_{ice} \Delta T \] \[ Q_1 = 3 \times 2100 \times 12 \] \[ Q_1 = 75600\,J \]

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Step 2: Melting Ice

\[ Q_2 = m L_f \] \[ Q_2 = 3 \times 3.35 \times 10^5 \] \[ Q_2 = 1.005 \times 10^6\,J \]

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Step 3: Heating Water to \(100^\circ C\)

\[ Q_3 = m s_w \Delta T \] \[ Q_3 = 3 \times 4186 \times 100 \] \[ Q_3 = 1.2558 \times 10^6\,J \]

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Step 4: Converting Water to Steam

\[ Q_4 = m L_v \] \[ Q_4 = 3 \times 2.256 \times 10^6 \] \[ Q_4 = 6.768 \times 10^6\,J \]

---

Total Heat Required

\[ Q = Q_1 + Q_2 + Q_3 + Q_4 \] \[ Q = 75600 + 1.005\times10^6 + 1.2558\times10^6 + 6.768\times10^6 \] \[ Q = 9.10 \times 10^6\,J \]

What is the temperature of the steel–copper junction in the steady state of the system shown in Fig. 10.15? Length of the steel rod = 15.0 cm, length of the copper rod = 10.0 cm, furnace temperature = \(300^\circ C\), temperature of the cold end = \(0^\circ C\). The cross-sectional area of the steel rod is twice that of the copper rod. (Thermal conductivity of steel \(= 50.2\,J\,s^{-1}m^{-1}K^{-1}\); thermal conductivity of copper \(= 385\,J\,s^{-1}m^{-1}K^{-1}\)).

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Concept / Theory

In a steady state of heat conduction, the temperature at every point of the system remains constant with time. This means the rate of heat flow through each section of the conductor is the same.

According to Fourier’s law of conduction, the rate of heat flow through a rod is given by

where \(K\) is the thermal conductivity, \(A\) is the cross-sectional area, and \(L\) is the length of the conductor. In steady state, heat current through the steel rod equals the heat current through the copper rod.

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Solution

Given:

• Length of steel rod \(L_s = 15.0\,cm = 0.15\,m\)
• Length of copper rod \(L_c = 10.0\,cm = 0.10\,m\)
• Hot end temperature \(T_H = 300^\circ C\)
• Cold end temperature \(T_C = 0^\circ C\)
• Area of steel rod \(A_s = 2A\)
• Area of copper rod \(A_c = A\)
• Thermal conductivity of steel \(K_s = 50.2\,J\,s^{-1}m^{-1}K^{-1}\)
• Thermal conductivity of copper \(K_c = 385\,J\,s^{-1}m^{-1}K^{-1}\)

Let the temperature at the steel–copper junction be \(T^\circ C\).

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Step 1: Heat Flow Through Steel Rod

\[ H_s = \frac{K_s A_s (300 - T)}{L_s} \] \[ H_s = \frac{50.2 \times 2A (300 - T)}{0.15} \]

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Step 2: Heat Flow Through Copper Rod

\[ H_c = \frac{K_c A_c (T - 0)}{L_c} \] \[ H_c = \frac{385 \times A \times T}{0.10} \]

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Step 3: Apply Steady-State Condition

In steady state, \[ H_s = H_c \] \[ \frac{50.2 \times 2A (300 - T)}{0.15} = \frac{385 \times A T}{0.10} \] Cancelling \(A\), \[ \frac{100.4(300 - T)}{0.15} = 3850T \] \[ 100.4(300 - T) = 577.5T \]

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Step 4: Solve for Junction Temperature

\[ 100.4 \times 300 - 100.4T = 577.5T \] \[ 30120 = 677.9T \] \[ T = \frac{30120}{677.9} \] \[ T \approx 44.4^\circ C \]

An iron bar \((L_1 = 0.1\,m,\ A_1 = 0.02\,m^2,\ K_1 = 79\,W\,m^{-1}K^{-1})\) and a brass bar \((L_2 = 0.1\,m,\ A_2 = 0.02\,m^2,\ K_2 = 109\,W\,m^{-1}K^{-1})\) are soldered end to end as shown in Fig. 10.16. The free ends of the iron and brass bars are maintained at \(373\,K\) and \(273\,K\) respectively. Determine:
(i) the temperature of the junction,
(ii) the equivalent thermal conductivity of the compound bar,
(iii) the heat current through the bar.

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Concept / Theory

When two conducting rods are joined end-to-end and heat flows through them, the system eventually reaches a steady state. In steady state, the rate of heat flow through each section of the rod is the same.

According to Fourier’s law of heat conduction:

For rods connected in series, the heat current is equal through both rods. This condition allows us to determine the temperature at the junction.

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Solution

Given:

• Iron rod: \(L_1=0.1\,m,\ A_1=0.02\,m^2,\ K_1=79\,W\,m^{-1}K^{-1}\)
• Brass rod: \(L_2=0.1\,m,\ A_2=0.02\,m^2,\ K_2=109\,W\,m^{-1}K^{-1}\)
• Hot end temperature \(T_1=373\,K\)
• Cold end temperature \(T_2=273\,K\)

Let the temperature at the junction be \(T_0\).

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1️⃣ Junction Temperature

In steady state: \[ \frac{K_1 A_1 (373-T_0)}{L_1} = \frac{K_2 A_2 (T_0-273)}{L_2} \] Since \(A_1=A_2\) and \(L_1=L_2\), \[ K_1(373-T_0)=K_2(T_0-273) \] \[ 79(373-T_0)=109(T_0-273) \] \[ 79\times373-79T_0=109T_0-109\times273 \] \[ 29467+29757=188T_0 \] \[ 59224=188T_0 \] \[ T_0=\frac{59224}{188} \] \[ T_0 \approx 315\,K \]

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2️⃣ Equivalent Thermal Conductivity

For two rods in series with equal length and area: \[ K_{eq}=\frac{2K_1K_2}{K_1+K_2} \] \[ K_{eq}=\frac{2\times79\times109}{79+109} \] \[ K_{eq}=\frac{17222}{188} \] \[ K_{eq}\approx91.6\,W\,m^{-1}K^{-1} \]

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3️⃣ Heat Current Through the Bar

Total length of the compound bar: \[ L=0.1+0.1=0.2\,m \] Using Fourier’s law, \[ H=\frac{K_{eq}A(T_1-T_2)}{L} \] \[ H=\frac{91.6\times0.02\times(373-273)}{0.2} \] \[ H=\frac{91.6\times0.02\times100}{0.2} \] \[ H=\frac{183.2}{0.2} \] \[ H=916\,W \]

A pan filled with hot food cools from \(94^\circ C\) to \(86^\circ C\) in 2 minutes when the room temperature is \(20^\circ C\). How long will it take to cool from \(71^\circ C\) to \(69^\circ C\)?

---

Concept / Theory

This problem is based on Newton’s Law of Cooling, which states that the rate of loss of heat of a body is proportional to the difference between the temperature of the body and that of the surroundings.

For small temperature intervals, the cooling rate can be approximated as proportional to the average temperature excess above the surroundings.

---

Solution

Given:

• Cooling from \(94^\circ C\) to \(86^\circ C\)
• Time taken \(= 2\ \text{min} = 120\ s\)
• Room temperature \(T_0 = 20^\circ C\)
• Required cooling: \(71^\circ C\) to \(69^\circ C\)

---

Step 1: First Cooling Interval

Temperature drop: \[ \Delta \theta_1 = 94 - 86 = 8^\circ C \] Average temperature: \[ \frac{94 + 86}{2} = 90^\circ C \] Average temperature excess above surroundings: \[ \Delta T_1 = 90 - 20 = 70^\circ C \] Cooling rate relation: \[ \frac{\Delta \theta_1}{\Delta t_1} = k' \Delta T_1 \] \[ \frac{8}{120} = k' \times 70 \] \[ k' = \frac{8}{120 \times 70} \]

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Step 2: Second Cooling Interval

Temperature drop: \[ \Delta \theta_2 = 71 - 69 = 2^\circ C \] Average temperature: \[ \frac{71 + 69}{2} = 70^\circ C \] Average temperature excess: \[ \Delta T_2 = 70 - 20 = 50^\circ C \] Cooling relation: \[ \frac{\Delta \theta_2}{t} = k' \Delta T_2 \] Substitute \(k'\): \[ \frac{2}{t} = \frac{8}{120 \times 70} \times 50 \]

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Step 3: Solve for Time

\[ \frac{2}{t} = \frac{8 \times 50}{120 \times 70} \] \[ t = \frac{2 \times 120 \times 70}{8 \times 50} \] \[ t = \frac{16800}{400} \] \[ t = 42\ s \]