Thermodynamics: Process & Laws

Thermodynamics is a branch of physics that is closely related to the working mechanism of a system and energy. Thermodynamics was discovered by scientists back in the 19th century with the objective to develop steam engines. It majorly deals with the process of tracking the transfer of heat from one body to the other and its impact on the body. The approach of thermodynamics contributes towards understanding the change in state caused due to the transfer of heat. Follow this article to understand the basics of thermodynamics better.

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Thermodynamics: Zeroth Law

Zeroth law of thermodynamics states that if system \(A\) is in thermal equilibrium with system \(C\) and system \(B\) is in thermal equilibrium with system \(C\) then system \(A\) and system \(B\) will also be in equilibrium with each other.

Thermodynamics: First Law

The first law of thermodynamics is an extension of law of conservation of energy, and it further states that for a closed system, a change in internal energy is equal to the difference of the heat supplied to the system and work done by the system.

In other words, the heat supplied to a system is equal to the sum of the change in the internal energy of the system and work done by the system.

Thermodynamics: Second Law

The second law of thermodynamics states that the entropy or the randomness of the universe is always increasing.
In other words, for any cyclic process, even if all the processes are ideal and the energy lost is negligible, we cannot achieve a complete hundred percent efficiency.

System

The object or matter with boundary and defined permeabilities. Our discussion and study involve all the matter inside the system.
Surrounding: All the space that exists beyond the boundary of the system is known as surrounding.
Universe: The system together with its surroundings is known as the Universe.

Open system is a type of system which allows both the transfer of energy and mass.
Closed system is a type of system which only allows the transfer of energy and not mass.
Isolated system is a type of system which doesn’t allow the transfer of both mass and energy.

System Variables

The thermodynamics state of a system during a thermodynamic process is defined by some variables like temperature, pressure, volume etc. These variables are known as system variables.
System variables can be used to pictorially represent a thermodynamic process.
Graphical representation of a thermodynamical process is known as Indicator Diagram.

For a gas piston closed system, there are three system variables that are required to define the state of the system, i.e., Pressure, Temperature and Volume.

Also, if we consider Ideal Gas Law, effectively two parameters can completely define the state of the Gas-piston system.

Isothermal Process

Isothermal Process is a thermodynamic process in which the temperature remains constant throughout the process from the initial to the final state.
\(T = \rm{const}\)
Since, the temperature is constant,
\(∆ T = 0\)
Internal energy is given by,
\(nC_v ∆T = 0\)
\(∆U = 0 ⇒ U = \rm{const}\)
Equation of the Isothermal process is given by,
\(PV = \rm{const}\)
\(P_i V_i = P_f V_f\)
Work done is given by,
\({{\Delta }}W = \int_{{V_i}}^{{V_f}} {PdV} \)
From Ideal Gas Law,
\(P = \frac{{nRT}}{V}\)
\({{\Delta }}W = \int_{{V_i}}^{{V_f}} {\frac{{nRT}}{V}dV} \)
\(\Delta W = nRT{\text{ln}}\left({\frac{{{V_f}}}{{{V_i}}}} \right)\)
\(\Delta W = nRT{\text{ln}}\left({\frac{{{P_i}}}{{{P_f}}}} \right)\)
Molar Heat Capacity is for an isothermal process is given by,
\({C_{iso}} = \frac{\Delta Q}{nR{\Delta T}}\)
\(C_{iso} = \infty\)
Indicator Diagram

Isobaric Process

Isobaric Process is a thermodynamic process in which the pressure remains constant throughout the process from the initial to the final state.
\(P = \rm{const}\)
Internal energy is given by,
\(∆U = nC_v ∆T\)
From Ideal Gas Law,
\(P = \frac{{nRT}}{V}\)
\(P \propto \frac{T}{V}\)
Thus,
\(\frac{V}{T} = \rm{const}\)
\(\frac{{{V_i}}}{{{T_i}}} = \frac{{{V_f}}}{{{T_f}}}\)
The work done during the thermodynamic process is given by,
\(\Delta W = \int_{{V_i}}^{{V_f}} {PdV = P\left({{V_f} – {V_i}} \right) = P\Delta V}\)
\(∆ W = P ∆ V\)
Using the ideal gas law we have,
\(∆ W = nR ∆ T\)
Molar heat capacity at constant pressure is given by,
\({C_p} = \frac{{\left({\Delta Q}\right)}_p}{{n\Delta T}} = {\text{const}}\)
Indicator Diagram

Isochoric Process

Isochoric process is a thermodynamic process in which the volume of the system remains constant from the initial to the final state.
Thus, for an isochoric process we have,
\(V = \rm{const}\)
\(⇒ ∆V = 0\)
Internal energy is given by,
\(∆U = nC_v ∆T\)
Since there is no change in the volume, all the heat lost or gained goes into changing the internal energy.
\(V = \rm{const}\)
From ideal gas law,
\(V \propto \frac{T}{P}\)
\( \Rightarrow \frac{P}{T} = {\text{const}}\)
\( \Rightarrow \frac{{{P_i}}}{{{T_i}}} = \frac{{{P_f}}}{{{T_f}}}\)
Since the volume remains constant, i.e., there is no change in the volume, thus, work done will be zero.
\(\Delta W = \int_{{V_i}}^{{V_f}} {PdV = P\left({{V_f} – {V_i}} \right) = 0} \)
\(∆ W = 0\)
Molar heat capacity at constant volume is given by,
\({C_v} = \frac{\left({\Delta Q} \right)_v}{{n\Delta T}} = {\text{const}}\)
Indicator Diagram

Adiabatic Process

Adiabatic process is a thermodynamic process in which there is no exchange of heat between the system and the surrounding from the initial to the final state.
Since the heat exchange is zero
\(∆Q = \rm{const} = 0\)
Internal energy of the system is given by,
\(∆U = nC_v ∆T\)
Equation of thermodynamic process is given by,
\(PV^γ = \rm{const}\)
\(⇒ TV^{γ-1} = \rm{const}\)
Where,
\(\gamma = \frac{{{C_p}}}{{{C_v}}}\)
Work done is given by,
\(\Delta W = \int_{{V_i}}^{{V_f}} {PdV}\)
\( = \int_{{V_i}}^{{V_f}} {\frac{K}{{{V^\gamma }}}dV}\) and \(PV^γ = K\)
\( = K\left[{\frac{{{V^{ – \gamma + 1}}}}{{ – \gamma + 1}}} \right]_{{V_i}}^{{V_f}}\)
\(\frac{{KV_f^{ – \gamma + 1} – KV_i^{ – \gamma + 1}}}{{ – \gamma + 1}}\)
\( \Rightarrow \Delta W = \frac{{{P_f}{V_f} – {P_i}{V_i}}}{{1 – \gamma }}\)
\( \Rightarrow \Delta W = \frac{{nR\left({{T_i} – {T_f}} \right)}}{{1 – \gamma }}\)
Molar heat capacity for an adiabatic process is given by,
\(C = \frac{{\Delta Q}}{{n\Delta T}}\)
\(\Rightarrow C_\rm{adiabatic} = 0\)
Indicator Diagram

Polytropic Process

Polytropic process is a thermodynamic process whose equation of the process can be represented in the form of,
\(PV^n = \rm{const}\)
Where,
\(P\) is the pressure of the gas.
\(V\) is the volume of the gas.
\(n\) is known as polytropic index.
A polytropic process has a constant Molar Heat Capacity.
General Equation of a polytropic Process.
\(PV^n = \rm{const}\)
1. If the polytropic index is one, then the process is Isothermal
\(n = 1\)
\(PV = \rm{const}\)
2. If the polytropic index is zero, then the process is Isobaric
\(n = 0\)
\(PV = \rm{const}\)
3. If the polytropic index is gamma, then the process is Adiabatic
\(n = \gamma = \frac{{{C_p}}}{{{C_v}}}\)
\(PV^γ = \rm{const}\)
4. If the polytropic index is infinite, then the process is Isochoric
\(V = {\left({\frac{{{\text{const}}}}{P}}\right)^{\frac{1}{n}}}\)
\(n \to \infty \)
\(\Rightarrow V = \rm{const}\)
Molar heat capacity of a polytropic process is given by,
\(PV^n = \rm{const}\)
From first law of thermodynamics we have,
\(∆Q = ∆U + W\)
Now, dividing the equation with, \(∆T\), we get,
\(\frac{{\Delta Q}}{{\Delta T}} = \frac{{\Delta U}}{{\Delta T}} + \frac{{\Delta W}}{{\Delta T}}\)
\(\frac{{\Delta Q}}{{\Delta T}} = \frac{{{C_v}\Delta T}}{{\Delta T}} + \frac{{PdV}}{{\Delta T}}\)
From ideal gas law,
\(dT = PdV + VdP\)
\( \Rightarrow C = {C_v} + \frac{{RPdV}}{{PdV + VdP}}\)
From the equation of the process,
\(P.n{V^{n – 1}}dV + {V^n}dP = 0\)
\(dP = \frac{{ – nPdV}}{V}\)
\( \Rightarrow C = {C_v} + \frac{{RPdV}}{{PdV + V\left({\frac{{ – nPdV}}{V}} \right)}}\)
Therefore, the molar heat capacity is given by,
\( \Rightarrow C = {C_v} + \frac{R}{{1 – n}}\)

Application

Thermodynamic processes are used in engines.

Thermodynamic processes are used in thermal and nuclear power plants.

Sample Problem

Q.1. The following graph gives the different indicator diagrams representing the common thermodynamic process. Identify the possible thermodynamic process for each indicator diagram and give reasons.

Ans: For Isobaric process, the pressure remains constant; thus, in the above graph the curve \(1\) is parallel to horizontal and has constant pressure thus, curve \(1\) is isobaric,
For isochoric process we have the volume that remains constant; thus, in the above graph the curve \(6\) is vertical thus, volume remains constant along the curve thus, the curve \(6\) represents an isochoric process.
Since it is said that the curves represent an important polytropic process thus, the curve \(2,\,3,\,4,\,5\) will be represented by \(PV^n = k\)
As the value of \(n\) increases, the curve occupies the lower position along the vertical.
We know that for \(n = 1\) the curve will be isothermal and since the polytropic indexes for the adiabatic process are greater than \(1\) thus curve \(2\) will be isothermal.
Now, for different atomicity of the gas, the value of gamma for the adiabatic process will change.
The gamma is inversely proportional to the atomicity of the gas; thus, the gas with the lowest atomicity, i.e., adiabatic process for monoatomic gas will occupy the lowest position along the vertical.
Therefore, curve \(3\) represents the adiabatic process for a polyatomic gas, curve \(4\) represents the adiabatic process for diatomic gas, and curve \(5\) represents curve for the adiabatic process for monoatomic gas.

Q.2. Following is the Density-Temperature curve for a cyclic process. Find the heat exchanged, Work done and change in internal energy for the three processes involved in the cyclic process.

Ans: For the process \(AB\), since the system is a closed system thus, the process will be isochoric if the density remains constant,
Thus,
The work done will be zero, i.e., \(∆W_{AB} = 0\)
The change in internal energy is given by,
\(∆U_{AB} = nC_v ∆T_{AB}\)
\( \Rightarrow {{\Delta }}{U_{{{AB}}}} = \frac{3}{2}R\left({ – 2{T_0}} \right)\)
Heat transfer in the process is given by,
\(∆Q_{AB} = ∆U_{AB} + ∆W_{AB}\)
\(⇒ ∆Q_{AB} = – 3RT_0\)
For the process \(BC\),
\(ρ = kT\)
\(⇒ \frac{M}{V} = kT\)
Substituting temperature from ideal gas laws we get,
\(PV^2 = k\)
Thus, the polytropic index is equal to two.
\(n = 2\)
Work done for a polytropic process is given by,
\({{\Delta }}W = \frac{{nR\left({{T_i} – {T_f}} \right)}}{{n – 1}}\)
\({{\Delta }}{W_{{{BC}}}} = \frac{{R\left({{T_i} – {T_f}} \right)}}{{n – 1}} = 2R{T_0}\)
The change in internal energy is given by,
\(∆U_{BC} = nC_v ∆T_{BC}\)
\(∆U_{BC} = 3RT_0\)
Heat transfer in the process is given by,
\(∆Q_{BC} = ∆U_{BC} + ∆W_{BC}\)
\(∆Q_{BC} = 5RT_0\)
For the process \(CA\), since the temperature remains constant the process is isothermal.
Work done is given by,
\({{\Delta }}{W_{{{CA}}}} = RT\ln \left({\frac{{{V_f}}}{{{V_i}}}}\right)\)
\(⇒ ∆W_{CA} = 3RT_0 \ln(3)\)
Since the process is isothermal, the temperature remains constant; thus, the change in internal energy is zero.
\(⇒ ∆U_{CA} = 0\)
Heat exchange is given by,
\(∆Q_{CA} = ∆U_{CA} + ∆W_{CA}\)
\(⇒ ∆Q_{CA} = 3RT_0 \ln(3)\)

Summary

Thermodynamic process is defined as the constraints that the system variables undergo while the system transitions from the initial to the final state.
For a gas-piston system, pressure, temperature, and volume are important system variables.
Polytropic processes are thermodynamic processes which has constant molar heat capacity.
Non polytropic process has variable molar heat capacity.
Change in internal energy for any thermodynamic process is given by,
\(∆U = nC_v ∆T\)
Thermodynamic process finds its application in a number of areas, for example, thermal power plants, refrigerators, diesel, and petrol engines.

Frequently Asked Questions on Thermodynamic Process

Frequently asked questions related to the process of thermodynamics is listed as follows:

Q. What is the molar heat capacity for an isothermal process?
Ans: For an isothermal process the temperature remains constant since molar heat capacity is inversely proportional to the change in temperature; thus, the molar heat capacity is infinite.

Q. What is an isothermal process?
Ans: Isothermal process is a thermodynamic process in which the temperature remains constant.

Q. What is adiabatic process?
Ans: Adiabatic process is a thermal process in which there is no exchange of heat between the system and surroundings, and the work is done at the expense of internal energy.

Q. What is the change in internal energy in a cyclic process?
Ans: Internal Energy is a state function therefore in the cyclic process the initial and the final state of the system is same thus the internal energy in final and initial state remains same thus the change in internal energy is zero in a cyclic process.

Q. What is the polytropic index for adiabatic process?
Ans: For adiabatic process, the polytropic index is gamma, whose value depends on the atomicity of the gas in the system.

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