Question

Derive the work done in an adiabatic process.

Answer 1

Work done in an adiabatic process: Consider µ moles of an ideal gas enclosed in a cylinder having perfectly non conducting walls and base. A frictionless and insulating piston of cross sectional area A is fitted in the cylinder. 

Let W be the work done when the system goes from the initial state (P_i, V_i, T_i) to the final state (P_f, V_f ,T_f) adiabatically.

By assuming that the adiabatic process occurs quasi-statically, at every stage the ideal gas law is valid. Under this condition, the adiabatic equation of state is PV_γ = constant (or)

In adiabatic expansion, work is done by the gas. i.e., W_adia is positive. As T_i > T_f the gas cools during adiabatic expansion. 

In adiabatic compression, work is done on the gas. i.e., W_adia is negative. As T_i < T_f the temperature of the gas increases during adiabatic compression.

PV diagram- work done in the adiabatic process

To differentiate between isothermal and adiabatic curves in PV diagram, the adiabatic curve is drawn along with isothermal curve for T_f and T_i. 

Note that adiabatic curve is steeper than isothermal curve. This is because γ > 1 always.