Workdone during adiabatic expansion is given by

where p₁, v₁, T₁ = Pressure, volume and temperature for the initial condition of gas,

p₂, v₂, T₂ = Corresponding values for the final condition of gas,

R = Gas constant, and

γ = Ratio of specific heats

Explanation:

Adiabatic process:

• It is a thermodynamic process in which there is no exchange of heat from the system to its surroundings neither during expansion nor during compression, yet the temperature may change.
• For the adiabatic process to occur, the system must be perfectly insulated from the surrounding.
• A reversible adiabatic process is called isentropic.

For the adiabatic process, the work done by the system is given as

\(W = \frac{{{P_1}{V_1} - {P_2}{V_2}}}{{\gamma - 1}} = \frac{{nR\left( {{T_1} - {T_2}} \right)}}{{\gamma - 1}}\)

\(W = \frac{p_1v_1}{\gamma -1}\left(1-\frac{p_2v_2}{p_1v_1}\right)\)

\(\Rightarrow W = \frac{mRT_1}{\gamma -1}\left(1-\frac{p_2v_2}{p_1v_1}\right)\)

where R is gas constant, n is molar mass and, \(\gamma = \frac{{{C_P}}}{{{C_V}}}\) , where Cp and Cv are specific heat at constant pressure and volume

Additional Information 

Isochoric process: 

• It is a thermodynamic process that takes place at constant volume.
• It is also known as an isometric process.
• In such a process, the work done is zero. The volume of the gas remains constant.

For constant volume or Isochoric process work done is given by,

Since for isochoric process, we know that, dV = 0

So, W = 0 for the Isochoric process.

Isothermal process

• It is a thermodynamic process in which the temperature of a system remains constant.
• The thermal equilibrium is maintained because the heat transfer is very slow.
• The work done in this process is due to the change of net heat content of the system.

For the Isothermal process work is done given by,

\({W_{1 - 2}} = {p_1}{V_1}\ln \left( {\frac{{{p_1}}}{{{p_2}}}} \right) = {p_2}{V_2}\ln \left( {\frac{{{p_1}}}{{{p_2}}}} \right)\)

Isobaric process:

• It is a thermodynamic process in which the pressure remains constant.

For the Isobaric process, the work done is given by,

W = P ΔV

where P = pressure in N/m2, ΔV = Change in volume.