Write expression for the work done by 1 mole of the gas in each of the following cases:

(i) For irreversible expansion of the gas from volume V₁ to V₂

(ii) For reversible isothermal expansion of  the gas from volume V₁ to V₂

(iii) For expansion of the gas into an evaluated vessel.

(iv) For reversible isothermal compression of the gas from pressure P₁ to P₂.

For adiabatic expansion resulting into change of temperature from T₁ toT₂.

(i) Irreversible expansion takes place when external pressure (P_ext) remains constant.
W_irev = −P_ext(V₂ − V₁) = −P_ext ΔV

(ii) Reversible expansion takes place when internal pressure is infinitesimally greater than external pressure (P_int ≃ P_ext) at every state. Thus, external pressure has to be adjusted throughout.
w_rev = −nRT ln V₂ / V₁ 

(iii) The expansion is irreversible. Further, as P_ext = 0, therefore,
w = −P_ext ΔV = 0 × (ΔV) = 0

(iv) When gas is compressed, work is done on the gas. For isothermal reversible compression,
w = +nRT ln (P₂ / P₁)

(v) For adiabatic expansion, temperature falls, i.e., T₂ < T₁.
w = C_v(T₂ − T₁), i.e., w is −ve.