THERMODYNAMICS-Notes
Physics - Notes
THERMAL EQUILIBRIUM
Thermal equilibrium is the physical state of a system in which no net heat flow occurs within the system or between the system and its surroundings, even when the system is allowed to interact thermally. In this state, the temperature is uniform throughout the system and remains constant with time.
In simpler terms, when two or more bodies are placed in thermal contact and their macroscopic thermal properties stop changing, the system is said to have attained thermal equilibrium.
Physical Interpretation
Temperature is a measure of the degree of hotness of a body and determines the direction of heat flow. Heat flows spontaneously from higher temperature to lower temperature. Thermal equilibrium is achieved when this driving force vanishes, meaning:
\[\Delta T=0\]Once this condition is satisfied, the system has no reason to exchange heat further.
ZEROTH LAW OF THERMODYNAMICS
The Zeroth Law of Thermodynamics states that:
If two systems are each in thermal equilibrium with a third system, then the two systems are also in thermal equilibrium with each other.
This law provides the fundamental basis for the concept of temperature, even though it was formulated after the First and Second Laws.
Physical Meaning of the Zeroth Law
The Zeroth Law tells us that thermal equilibrium is a transitive relation. If two bodies independently show no heat exchange with a third body, they must share the same thermal condition.
This implies the existence of a single physical property—called temperature—that is:
- Identical for systems in thermal equilibrium
- Independent of the nature, size, or material of the systems
Derivation of Temperature Concept from the Zeroth Law
Let a physical quantity Texist such that:
\(T_A=T_C\) when A is in thermal equilibrium with C
\(T_B=T_C\) when B is in thermal equilibrium with C
Then it follows directly:
\[T_A=T_B\]
This quantity T is identified as temperature.
Thus, temperature is not defined by heat, but by the condition of thermal equilibrium.
Importance of the Zeroth Law
- Defines the concept of temperature
- Establishes temperature as a measurable quantity
- Enables construction and calibration of thermometers
- Provides the foundation for all thermal laws
- Separates temperature from heat and internal energy
The Zeroth Law is fundamental because it:
Without this law, temperature would be an undefined idea, and thermodynamics would lose its experimental meaning.
HEAT
Heat is the energy that flows from one system to another solely because of a difference in temperature. It is not a property contained within a body; instead, it represents energy in transit during a thermal interaction.
INTERNAL ENERGY
Internal energy is the total microscopic energy possessed by a system due to the random motion and interactions of its constituent particles.
It includes:
- Kinetic energy of molecules due to translation, rotation, and vibration
- Potential energy arising from intermolecular forces
WORK
Work in thermodynamics is the energy transferred to or from a system when an external force causes a macroscopic displacement of the system’s boundary.
FIRST LAW OF THERMODYNAMICS
The First Law of Thermodynamics states that:
The change in internal energy of a system is equal to the heat supplied to the system minus the work
done by the
system on its surroundings.
Mathematically, it is expressed as:
where
\(\Delta U\) = change in internal energy,
\(Q\) = heat supplied to the system,
\(W\) = work done by the system.
Derivation of the First Law
Internal energy \(U\) of a system can change through two modes of energy transfer : heat and work.
Let
\(\Delta Q\) = Heat supplied to the system by the surroundings
\(\Delta W\) = Work done by the system on the surroundings
\(\Delta U\) = Change in internal energy of the system
The general principle of conservation of energy then implies that
\[\Delta Q = \Delta U + \Delta W\]
the energy (∆Q) supplied to the system goes in partly to increase the internal energy of the
system \((\Delta U)\) and the rest in work on the environment \((\Delta W)\). Equation is known as the First
Law of Thermodynamics.
Limitations of the First Law
- The direction of heat flow
- Why all heat cannot be converted into work
While the First Law explains how much energy changes, it does not explain:
These aspects are addressed by the Second Law of Thermodynamics.
Important Aspects and Applications
- Establishes energy conservation in thermal systems
- Connects heat, work, and internal energy quantitatively
- Valid for all thermodynamic processes
- Foundation for engines, refrigerators, and energy analysis
- Essential for solving NCERT and competitive exam problems
SPECIFIC HEAT CAPACITY
Specific heat capacity of a substance is the amount of heat required to raise the temperature of unit mass of the substance by one degree, without any change in its physical state.
Derivation of the Formula
Suppose an amount of heat \(\Delta Q\) supplied to a substance changes its temperature from \(T\) to
\(T + \Delta T\). We define heat capacity of a substance
\[S=\dfrac{\Delta Q}{\Delta T}\]
Heat capacity \(S\) to be proportional to the mass of the substance.
Further, it could also depend on the temperature, i.e., a different amount of heat may
be needed for a unit rise in temperature at different temperatures. To define a constant
characteristic of the substance and independent of its amount, we divide \(S\) by the mass of the substance
\(m\) in kg :
Where,
\(s\) is known as the specific heat capacity of the substance.
The unit of specific heat capacity is \(\mathrm{J\ kg^{–1}\ K^{–1}}\).
If the amount of substance is specified in terms of moles \(\mu\) (instead of mass \(m\) in kg), we can define heat capacity per mole of the substance by
\[C=\dfrac{S}{\mu}=\dfrac{1}{\mu}\dfrac{\Delta Q}{\Delta T}\]\(C\) is known as molar specific heat capacity of the substance. The unit of \(C\) is \(\mathrm{J\ mo1^{–1}\ K^{–1}}\).
Specific heat capacity of water
The old unit of heat was the calorie. One calorie was originally defined as the amount of heat required to raise the temperature of 1 g of water by 1 °C. This definition was adopted because water is easily available and shows reproducible thermal behaviour.
With the improvement of experimental methods, it was found that the specific heat capacity of water does not remain exactly constant, but varies slightly with temperature. Careful calorimetric measurements reveal that the amount of heat required to raise the temperature of water by 1 °C depends on the initial temperature of the water. This variation is observed over the temperature range from 0 °C to 100 °C, as shown in Fig. 11.5.
Because of this temperature dependence, the original definition of the calorie was not sufficiently precise. To remove ambiguity, it became necessary to specify a definite temperature interval. Accordingly, one calorie is now defined as the amount of heat required to raise the temperature of 1 g of water from 14.5 °C to 15.5 °C under standard conditions.
Since heat is merely a form of energy transfer, it is conceptually preferable to use the same unit for heat, work, and all other forms of energy. The SI unit of energy is the joule (J). In SI units, the specific heat capacity of water is
\[\begin{aligned} c_{\text{water}} &= 4186\ \text{J kg}^{-1}\text{K}^{-1} \&= 4.186\ \text{J g}^{-1}\text{K}^{-1} \end{aligned}\]
The quantity historically known as the mechanical equivalent of heat was defined as the amount of mechanical work required to produce 1 calorie of heat. In modern terms, this quantity is simply a numerical conversion factor between two different units of energy, namely the calorie and the joule. Since the SI system uses the joule uniformly for heat, work, and internal energy, the term “mechanical equivalent of heat” has lost its independent physical significance and is no longer required.
An important feature of specific heat capacity is that it depends on the process or the conditions under which heat transfer takes place. This dependence becomes especially clear in the case of gases, for which two different specific heat capacities are defined: one at constant volume and the other at constant pressure.
For an ideal gas, the molar specific heat capacities at constant pressure \(C_p\) and at constant volume \(C_v\) are related by the simple relation
\[ C_p - C_v = R \]
where \(R\) is the universal gas constant. To prove this relation, we begin with the first law of thermodynamics applied to one mole of an ideal gas,
\[ \Delta Q = \Delta U + P\,\Delta V \]
If heat is absorbed at constant volume, then \(\Delta V = 0\), and hence
\[ \Delta Q_v = \Delta U \]
The molar specific heat capacity at constant volume is therefore
\[C_v = \left( \dfrac{\Delta Q}{\Delta T} \right)_v = \left( \dfrac{\Delta U}{\Delta T} \right)\tag{1} \]
If, on the other hand, heat is absorbed at constant pressure, the gas expands and does work. In this case,
\[ \Delta Q_p = \Delta U + P\,\Delta V \]
Dividing throughout by \(\Delta T\), the molar specific heat capacity at constant pressure becomes
\[ C_p = \left( \dfrac{\Delta U}{\Delta T} \right) + P \left( \dfrac{\Delta V}{\Delta T} \right) \tag{2}\]
For one mole of an ideal gas, the equation of state is
\[ PV = RT \]
At constant pressure, this gives
\[ P \left( \dfrac{\Delta V}{\Delta T} \right) = R \]
Substituting this result into Eq. (2) and using Eq. (1), we obtain
\[ C_p = C_v + R \]
which immediately leads to the required relation
\[\boxed{\; C_p - C_v = R\;} \]
THERMODYNAMIC STATE VARIABLES AND EQUATION OF STATE
A thermodynamic state variable is a physical quantity whose value depends only on the present state of the system and not on the path by which the system has reached that state.
Common examples of state variables include pressure, volume, temperature, internal energy, entropy, and enthalpy. These quantities are measurable or can be calculated from measurable quantities, and they uniquely characterize the equilibrium state of a system.
A defining feature of state variables is that their change between two states depends only on the initial and final states, not on the process connecting them.
Classification of State Variables
Thermodynamic state variables are broadly classified into two categories:
- Extensive Variables:
These variables depend on the size or amount of matter in the system. Examples include mass, volume, internal energy, and entropy. If the system is divided into two equal parts, the value of each extensive variable also gets divided equally. - Intensive Variables:
These variables are independent of the amount of matter in the system. Pressure, temperature, and density fall into this category. Dividing the system does not alter the value of an intensive variable.
This distinction is fundamental in thermodynamics and plays a key role in formulating equations of state and thermodynamic laws.
Thermodynamic Equilibrium and State Description
- Mechanical equilibrium:
(no unbalanced forces; pressure is uniform), - Thermal equilibrium:
(uniform temperature), - Chemical equilibrium:
(no net chemical reactions or diffusion).
A system is said to be in thermodynamic equilibrium when it simultaneously satisfies:
Only when these conditions are satisfied can the state of the system be uniquely described by state variables. In non-equilibrium situations, state variables may not be well defined.
Derivation of the Ideal Gas Equation of State (Conceptual)
Experimental studies show that for a fixed amount of gas:
\[\dfrac{PV}{T}=\text{constant}\]This constant depends on the quantity of gas present. For one mole of gas, it is denoted by \(R\), leading to
\[PV=RT\]For \(\mu\) moles
\[\boxed{\;PV=\mu RT\;}\]Important Physical Aspects and Significance
Thermodynamic state variables provide a bridge between microscopic molecular motion and macroscopic physical behaviour. The equation of state acts as a constraint that links these variables and reduces the number of independent quantities required to describe a system.
The concept of state variables is essential for understanding:
- the first law of thermodynamics,
- thermodynamic processes,
- heat engines and refrigerators.
The equation of state serves as the mathematical foundation on which these ideas are built.
Quasi-static Process
A quasi-static process is a thermodynamic process that proceeds so slowly that the system remains in thermodynamic equilibrium at every intermediate stage of the process.
In such a process, although the system is continuously changing, it passes through a succession of equilibrium states. As a result, all thermodynamic state variables such as pressure, volume, and temperature are well defined throughout the process.
Physical Meaning and Idealised Nature
The term “quasi-static” literally means “almost static.” In reality, no process can be perfectly quasi-static, because it would require an infinite amount of time to complete. Nevertheless, many real processes can be approximated as quasi-static if they are carried out sufficiently slowly.
For example, the slow compression or expansion of a gas using a piston, where external pressure is adjusted in extremely small steps, closely resembles a quasi-static process.
Quasi-static Process and Thermodynamic Equilibrium
- mechanical equilibrium is maintained by infinitesimal pressure differences,
- thermal equilibrium is preserved due to negligible temperature gradients,
- chemical equilibrium is unaffected if no reactions occur.
A quasi-static process ensures that the system remains in equilibrium because:
Isothermal Process
An isothermal process is a thermodynamic process in which the temperature of the system remains constant throughout the process.
Mathematically,
\[\Delta T=0\]For an isothermal process to occur, the system must be in good thermal contact with a large heat reservoir so that any heat lost or gained is immediately compensated, preventing any change in temperature.
Conditions for an Isothermal Process
For a process to be truly isothermal, it must proceed quasi-statically and sufficiently slowly. This ensures that thermal equilibrium between the system and its surroundings is maintained at every stage. Rapid processes generally do not allow enough time for heat exchange and therefore cannot be isothermal.
Derivation of Work Done in an Isothermal Process
Consider nmoles of an ideal gas undergoing a quasi-static isothermal expansion from volume \(V_1\) to \(V_2\) at temperature \(T\).
At any intermediate stage, the pressure of the gas is
\[P=\frac{\mu RT}{V}\]The infinitesimal work done by the gas during a small expansion dV is
\[dW=P\,dV=\frac{\mu RT}{V} dV\]Integrating between the limits \(V_1\) and \(V_2\),
\[\begin{aligned} W&=\int\limits_{V_1}^{V_2}\frac{\mu RT}{V} dV\\\\ &=\mu RT\ln\left({\dfrac{V_2}{V_1}}\right) \end{aligned}\]This expression gives the work done by an ideal gas in a quasi-static isothermal process.
Proof That Internal Energy Remains Constant
For an ideal gas, the internal energy depends only on temperature. Since temperature remains constant during an isothermal process,
\[\Delta U=0\]From the first law of thermodynamics,
\[\Delta Q=\Delta U+W\]Substituting \(\Delta U=0\),
\[\Delta Q=W\]This result proves that in an isothermal process, the heat absorbed by the gas is entirely converted into work done by the gas, with no change in internal energy.
Heat Exchange in Isothermal Compression and Expansion
During isothermal expansion, the gas does work on the surroundings. To maintain constant temperature, an equal amount of heat must flow into the gas from the surroundings.
During isothermal compression, work is done on the gas. The gas releases an equal amount of heat to the surroundings to keep its temperature unchanged.
Thus, continuous heat exchange with the surroundings is an essential feature of an isothermal process.
Adiabatic process
An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings.
Mathematically,
\[\Delta Q=0\]An adiabatic process can be realised either by perfect thermal insulation of the system or by carrying out the process very rapidly, so that there is insufficient time for heat transfer.
Derivation of Work Done in an Adiabatic Process
For an adiabatic process of an ideal gas.
\[PV^\gamma=\text{constant}\]where \(\gamma\) is the ratio of specific heats (ordinary or molar) at constant pressure and at constant volume.
\[\displaystyle \gamma = \dfrac{C_p}{C_p}\]Thus if an ideal gas undergoes a change in its state adiabatically from \((P_1,\ V_1 )\) to \((P_2,\ V_2 )\) :
\[P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \tag{1}\]We can calculate, as before, the work done in an adiabatic change of an ideal gas from the state \((P_1,\ V_1,\ T_1 )\) to the state \((P_2,\ V_2,\ T_2 )\).
\[ \begin{aligned} W &= \int_\limits{V_1}^{V_2} P \, dV \\\\ &= \text{constant} \times \int_\limits{V_1}^{V_2} \frac{dV}{V^{\gamma}} \\\\ &= \text{constant} \times \left[ \frac{V^{1 - \gamma}}{1 - \gamma} \right]_{V_1}^{V_2}\\\\ &= \text{constant} \times \left[\dfrac{1}{V_2^{\gamma -1}}-\dfrac{1}{V_1^{\gamma -1}} \right] \end{aligned} \]From Eq. (1), the constant is \(P_1 V_1^γ\text{ or }P_2 V_2^γ\)
\[\begin{aligned}W&=\dfrac{1}{1-\gamma}\left[ \dfrac{P_2V_2^{\gamma}}{V_2^{\gamma -1}}-\dfrac{P_1V_1^{\gamma}}{V_1^{\gamma}} \right]\\\\ &=\dfrac{1}{1-\gamma}\left[P_2V_2-P_1V_1\right]\\\\ &=\dfrac{\mu R(T_1-T_2)}{\gamma -1} \end{aligned} \]Comparison with Isothermal Process
On a P–V diagram, an adiabatic curve is steeper than the corresponding isothermal curve passing through the same initial point. This is because, in an adiabatic expansion, the temperature falls, causing pressure to drop more rapidly with increasing volume.
This comparison reinforces the distinct physical nature of adiabatic and isothermal processes.
Isochoric Process
An isochoric process is a thermodynamic process in which the volume of the system remains constant throughout the process.
Mathematically,
\[\Delta V=0\]Such a process typically occurs when a gas is enclosed in a rigid, non-expandable container, so that no mechanical expansion or compression is possible.
First Law Applied to an Isochoric Process
The first law of thermodynamics is given by
\[\Delta Q=\Delta U+W\]For an isochoric process, since \(\Delta V=0\),
\[W=\int P\,dV=0\]Hence, the first law reduces to
\[\Delta Q=\Delta U\]This relation shows that in an isochoric process, all the heat supplied to the system goes into changing its internal energy, with no work done by or on the system.
Proof That No Work Is Done in an Isochoric Process
Work done in a thermodynamic process is given by
\[W=\int P\, dV\]Since the volume remains constant, \(dV=0\) at every stage of the process. Therefore, \(W=0\).
This proves that mechanical work cannot be performed in an isochoric process, irrespective of changes in pressure or temperature.
Second Law of Thermodynamics
The first law of thermodynamics establishes the principle of energy conservation, but it does not indicate the direction in which thermodynamic processes occur. It places no restriction on whether heat can be completely converted into work or whether a process can spontaneously reverse itself. These limitations are provided by the second law of thermodynamics, which introduces the concept of irreversibility and directionality in natural processes.
The second law explains why certain processes occur naturally while their reverse never takes place without external intervention
Kelvin-Planck statement
The Kelvin–Planck statement may be expressed as follows:
It is impossible to construct a device that operates in a cyclic process and produces no effect other than the absorption of heat from a single thermal reservoir and the complete conversion of that heat into work.
In simpler terms, no heat engine can convert all the heat absorbed from a single reservoir entirely into work while operating in a cycle.
Clausius statement
The Clausius statement can be expressed as follows:
It is impossible to construct a device that operates in a cyclic process and has no effect other than the transfer of heat from a colder body to a hotter body.
In simpler terms, heat cannot flow spontaneously from a colder body to a hotter body without the aid of external work.
Reversible and Irreversible Processes
In thermodynamics, not all processes are equally feasible or symmetric in nature. Some processes, once completed, can be exactly reversed so that both the system and the surroundings return to their original states. Others cannot be undone without leaving permanent changes. This fundamental distinction leads to the classification of thermodynamic processes into reversible and irreversible processes. The study of these processes is essential for understanding the direction of natural phenomena and the limitations imposed by the second law of thermodynamics.
Reversible Process
A reversible process is a thermodynamic process that can be reversed by an infinitesimal change in a condition such that both the system and the surroundings are restored exactly to their initial states, with no net change anywhere in the universe.
In a reversible process, the system passes through a continuous sequence of equilibrium states, making the process an idealisation rather than a real physical occurrence.
Essential Conditions for Reversibility
- The process must be quasi-static, ensuring that the system remains in thermodynamic equilibrium at every stage.
- There must be no dissipative forces such as friction, viscosity, or turbulence.
- Heat transfer, if any, must occur across an infinitesimal temperature difference.
- There must be no unbalanced mechanical or chemical driving forces.
For a process to be reversible, the following conditions must be satisfied:
If any one of these conditions is violated, the process becomes irreversible.
Work Done in a Reversible Process (Conceptual Derivation)
In a reversible expansion of a gas, the external pressure differs from the internal gas pressure by an infinitesimally small amount. Hence, at every stage, pressure is well defined, and the work done is given by
\[W=\int P\, dV\]Because the system remains in equilibrium throughout, this expression gives the maximum possible work obtainable between two given states. This is a defining feature of reversible processes.
Irreversible Process
An irreversible process is a thermodynamic process that cannot be reversed without leaving permanent changes either in the system or in the surroundings.
Once such a process has occurred, restoring both the system and surroundings to their original states requires external intervention and results in net changes elsewhere.
Reversible vs Irreversible Process
| Basis of Comparison | Reversible Process | Irreversible Process |
|---|---|---|
| Definition | A process that can be reversed in such a way that both the system and the surroundings return exactly to their initial states. | A process that cannot be reversed without leaving permanent changes in the system or the surroundings. |
| Thermodynamic Equilibrium | The system remains in thermodynamic equilibrium at every stage (quasi-static process). | The system does not remain in equilibrium during intermediate stages. |
| Speed of Process | Infinitely slow and idealised. | Takes place in finite time. |
| Driving Force | Operates due to infinitesimal differences in pressure, temperature, or other state variables. | Operates due to finite differences in pressure, temperature, or other driving forces. |
| Dissipative Effects | Completely absent; no friction, viscosity, turbulence, or electrical resistance. | Always present; includes friction, viscosity, turbulence, electrical resistance, etc. |
| Work Done | Maximum possible work is obtained between two given states. | Work obtained is less than the maximum due to dissipative losses. |
| Heat Transfer | Takes place through an infinitesimal temperature difference \(\Delta T \to 0\). | Takes place through a finite temperature difference \(\Delta T \neq 0\). |
| Entropy Change of Universe | Zero change in entropy of the universe, \(\Delta S_{\text{universe}} = 0\). | Positive change in entropy of the universe, \(\Delta S_{\text{universe}} > 0\). |
| Graphical Representation | Can be represented by a well-defined path on a \(P\!-\!V\) or \(T\!-\!S\) diagram. | Cannot be represented by a single well-defined thermodynamic path. |
| Occurrence in Nature | Purely idealised; does not occur exactly in nature. | All natural and real processes are irreversible. |
| Examples | Ideal quasi-static isothermal or adiabatic expansion without friction. | Free expansion of gas, heat flow from hot to cold, frictional motion, mixing of gases. |
CARNOT ENGINE
A Carnot engine is an ideal heat engine that operates in a reversible cycle between two thermal reservoirs at fixed temperatures and converts heat into work with the maximum possible efficiency.
It operates between:
- a hot reservoir at absolute temperature \(T_H\),
- a cold reservoir at absolute temperature \(T_C\), where \(T_H\gt T_C\).
Construction and Working Substance
The Carnot engine consists of an ideal gas enclosed in a perfectly insulated cylinder fitted with a frictionless piston. The system can be brought into thermal contact alternately with the hot and cold reservoirs. All processes are carried out quasi-statically, ensuring reversibility at every stage.
The Carnot Cycle: Four Reversible Processes
The Carnot engine operates in a cycle consisting of four reversible processes, executed in a definite sequence.
- Reversible Isothermal Expansion at
\(T_H\):
The gas is placed in contact with the hot reservoir at temperature \(T_H\). It expands slowly, absorbing heat \(Q_H\) from the reservoir. Since the temperature remains constant, the internal energy of the gas does not change, and the absorbed heat is entirely converted into work. - Reversible Adiabatic Expansion:
The system is thermally insulated, and the gas continues to expand without exchanging heat. The temperature of the gas decreases from \(T_H\) to \(T_C\) as work is done at the expense of internal energy. - Reversible Isothermal Compression at
\(T_C\):
The gas is now placed in contact with the cold reservoir at temperature \(T_C\). It is compressed slowly, and heat \(Q_C \)is rejected to the cold reservoir while the temperature remains constant. - Reversible Adiabatic Compression:
The system is again insulated, and the gas is compressed adiabatically. Its temperature rises from \(T_C\) back to \(T_H\), restoring the gas to its initial state and completing the cycle.
Derivation of Efficiency of a Carnot Engine
The efficiency \etaof a heat engine is defined as
\[\eta=\frac{W}{Q_H}\]where \(W\) is the net work done per cycle.
For a cyclic process,
\[W=Q_H-Q_C\]Hence,
\[\eta=1-\frac{Q_C}{Q_H}\]For a reversible Carnot engine, it can be shown that
\[\frac{Q_C}{Q_H}=\frac{T_C}{T_H}\]Therefore, the efficiency of a Carnot engine is
\[\boxed{\;\eta=1-\frac{T_C}{T_H}\;}\]This expression shows that the efficiency depends only on the temperatures of the reservoirs and not on the nature of the working substance.
