Thermodynamic Process

What is called a Thermodynamic Process?

A process in which at least two or more state variables of a system got change, is called a thermodynamic process.

For example – Consider that, in a piston cylinder arrangement, pressure of a certain quantity of gas is increases by pressing the piston slowly, while the temperature of the gas is maintained constant.

In this compression process –

Pressure of the gas has changed and increases.
Volume of the gas has changed and decreases.
Temperature of the gas is unchanged and remains constant.

Pressure and volume both are the state functions which has got changed. Hence, the above process is a thermodynamic process.

Different thermodynamic process which are important for the subject of consideration are as follows –

Isochoric process or Isometric process.
Isobaric process or Isopiestic process.
Isothermal or Isotropic process.
Adiabatic process.

These processes are explained as follows –

Isochoric or Isometric Process

A thermodynamic process in which the volume of system remains constant is called an isochoric or isometric process.

The equation of state for an isochoric process is –

$V = \text {Constant}$

So, change in volume of the system is zero i.e. $\Delta V = 0$

Therefore, $\Delta W = P \Delta V = P \times 0 = 0$

Hence, work done in an isochoric process us zero.

From first law of thermodynamics, we get –

$\Delta Q = \Delta U + \Delta W$

Or, $\quad \Delta Q = \Delta U + 0 = \Delta U$

Hence, in an isochoric process, the entire heat change used to change the internal energy of the system due to change in its temperature.

The change in temperature can be determined by the equation –

$Q = n C_V \ \Delta T$

Isobaric or Isopiestic Process

A thermodynamic process which occurs at constant pressure is called an isobaric or isopiestic process.

In isobaric process the volume and temperature of the system change but the pressure remains constant.
The equation of state for an isobaric process is –

$P = \text {Constant}$

An isochoric process strictly obeys Charle’s law.

Work Done in Isobaric Process

Suppose, a gas expands at constant pressure from initial state $( V_1, \ T_1 )$ to final state $( V_2, \ T_2 )$

Amount of work done by the gas will be –

$W_{isobaric} = \int\limits_{V_1}^{V_2} P dV$

Or, $\quad W_{isobaric} = P \int\limits_{V_1}^{V_2} dV$

$= P \left ( V_2 - V_1 \right ) = n R \left ( T_2 - T_1 \right )$

Therefore, work done in an isobaric process is given by –

$W_{isobaric} = n R \left ( T_2 - T_1 \right )$ ……….. (1)

Isothermal or Isotropic Process

An isothermal process is a thermodynamic process in which the process occurs at constant temperature conditions. It is also called an isotropic process.

In isothermal process, the pressure and volume of the system change but the temperature remains constant.
The boundary of the system is consisting of conducting walls so that it allows flow of heat to maintain the system temperature constant.
The process occur very slowly so as to provide sufficient time for the exchange of heat from system to surrounding or vice versa.

The equation of state for an isothermal process is –

$PV =$ Constant.

Therefore, $\quad P_1 V_1 = P_2 V_2$

An isothermal process strictly obeys Boyle’s law.

Work Done in Isothermal Process

Consider $( n )$ moles of an ideal gas contained in a cylinder of cross sectional area A and having conducting walls and friction-less piston as shown in figure.

When piston moves up through a small distance $( dx )$ , the work done will be –

$\Delta W = P A dx = P dV$

Then total amount of work done by the gas, when it expands isothermally from initial state $( P_1, \ V_1 )$ to final state $( P_2, \ V_2 )$ will be –

$W_{iso} = \int\limits_{V_1}^{V_2} P dV$

But, for $( n )$ moles of a gas –

$P V = n R T$

Or, $\quad P = \left ( \frac { n R T }{ V } \right )$

Therefore, $\quad W_{iso} = \int\limits_{V_1}^{V_2} \left ( \frac { n R T }{ V } \right ) dV$

$= n R T \int\limits_{V_1}^{V_2} \left ( \frac {1}{V} \right ) dV$

$= n R T \left [ \ln V_2 - \ln V_1 \right ]$

$= n R T \ln \left ( \frac {V_2}{V_1} \right )$

But, $\quad P_1 V_1 = P_2 V_2$

Therefore, $\quad W_{iso} = 2.303 \ n R T \log \left ( \frac {V_2}{V_1} \right )$

$= 2.303 \ n R T \log \left ( \frac {P_1}{P_2} \right )$

Therefore, work done in an isothermal process is given by –

$W_{iso} = 2.303 \ n R T \log \left ( \frac {V_2}{V_1} \right )$ ……… (2)

$= 2.303 \ n R T \log \left ( \frac {P_1}{P_2} \right )$ ………. (3)

Adiabatic Process

An adiabatic process is a thermodynamic process in which the process occurs without exchange of heat from surroundings.

In an adiabatic process pressure, volume and temperature of the system change but heat exchange not occur.
The boundary of the system consisting of non-conducting walls so that it doesn’t allows flow of heat to and from surroundings.
The process should occur very fast so that heat does not get enough time for the exchange from system to surrounding or vice versa.

Work Done in Adiabatic Process

Consider $( n )$ moles of an ideal gas contained in a cylinder of cross sectional area $( A )$ and having perfectly insulated walls and friction-less piston as shown in figure.

Suppose, the gas expands adiabatically from initial state $( P_1, \ V_1, \ T_1 )$ to final state $( P_2, \ V_2, \ T_2 )$

Total work done by the gas will be –

$W_{adia} = \int\limits_{V_1}^{V_2} P dV$

But, for an ideal gas in adiabatic process –

$P V^{\gamma} = \text {Constant}$

Or, $\quad P = K V^{- \gamma}$

Therefore, work done in an adiabatic process –

W_{adia} = \int\limits_{V_1}^{V_2}  KV^{- \gamma} dV

$= K \int\limits_{V_1}^{V_2} V^{- \gamma} dV$

$= K \left [ \frac { V^{ 1 - \gamma }}{ 1 - \gamma } \right ]_{V_1}^{V_2}$

$= \left ( \frac {K}{1 - \gamma} \right ) \left [ V^{1 - \gamma}_{2} - V^{1 - \gamma}_{1} \right ]$

But, $\quad P_1V^{\gamma}_1 = P_2V^{\gamma}_2 = K$

Therefore, $\quad W_{adia} = \left ( \frac {1}{1 - \gamma} \right ) \left [ P_2V_2V^{1 - \gamma}_{2} - P_1V_1V^{1 - \gamma}_{1} \right ]$

$= \left ( \frac {1}{1 - \gamma} \right ) \left [ P_2V_2 - P_1V_1 \right ]$

$= \left ( \frac {1}{\gamma - 1} \right ) \left [ P_1V_1 - P_2V_2 \right ]$

Also, $\quad P_1V_1 = n R T_1 \quad$ and $\quad P_2V_2 = n R T_2$

Therefore, $\quad W_{adia} = \left ( \frac {1}{\gamma - 1} \right ) \left [ nRT_1 - nRT_2 \right ]$

$= \left ( \frac {nR}{\gamma - 1} \right ) \left [ T_1 - T_2 \right ]$

Therefore, work done in an adiabatic process is given by –

$W_{adia} = \left ( \frac {1}{\gamma - 1} \right ) \left [ P_1V_1 - P_2V_2 \right ]$ ……… (4)

$= \left ( \frac {nR}{\gamma - 1} \right ) \left [ T_1 - T_2 \right ]$ …….. (5)

PV, PT & VT Diagrams

For better understanding of difference between curves relating to $( P ), \ ( V )$ and $( T )$ in thermodynamic processes we have plotted on the same axes as shown below. These are –

Pressure verses Volume diagram.
Pressure verses Temperature diagram.
Volume verses Temperature diagram.

These diagrams are explained as below –

Pressure-Volume Diagram

Figure shows Pressure-Volume diagram of expansion of a gas.

Blue colored curve indicates $PV$ diagram of an isobaric process. It is a straight line parallel to the volume axis.
Green colored curve indicates $PV$ diagram of an isothermal process. It is a curve of increasing slope.
Red colored curve indicates $PV$ diagram of an adiabatic process. It is a curve of increasing slope.
Orange colored curve indicates $PV$ diagram of an isochoric process. It is a straight line parallel to pressure axis.

Pressure-Temperature Diagram

Figure shows Pressure-Temperature diagram of expansion of a gas.

Blue colored curve indicates $PT$ diagram of an isobaric process. It is a straight line parallel to the temperature axis.
Green colored curve indicates $PT$ diagram of an isothermal process. It is a straight line parallel to the pressure axis.
Red colored curve indicates $PT$ diagram of an adiabatic process. It is curve with decreasing slope.
Orange colored curve indicates $PT$ diagram of an isochoric process. It is a slanting straight line with constant slope.

Volume-Temperature Diagram

Figure shows Volume-Temperature diagram of expansion of a gas.

Blue colored curve indicates $VT$ diagram of an isobaric process.
It is a slanting straight line with constant slope.
Green colored curve indicates $VT$ diagram of an isothermal process. It is a straight line parallel to the volume axis.
Red colored curve indicates $VT$ diagram of an adiabatic process.
It is a curve of decreasing slope.
Orange colored curve indicates $VT$ diagram of an isochoric process. It is a straight line parallel to temperature axis.

See numerical problems based on this article.