# Isothermal expansion

From the first law of thermodynamics,

Î”

*E*=*q*+*w*For isothermal process, Î”

*E*= 0. Hence,*q*= â€“

*w*

In the case of isothermal expansion, work is done by the system at the cost of heat absorbed. The magnitude of

*q*or*w*depends on the manner in which the process of expansion is carried out, i.e., reversibly or irreversibly.Î”

*H*can be calculated as follows:Î”

*H*= Î”*E*+ Î”*n*_{g}RTFor isothermal process, Î”

*E*= 0 and Î”*T*= 0; thus, Î”*H*= 0.**Work done in reversible isothermal expansion**

Work done in reversible isothermal expansion is given by

At constant temperature,

âˆ´

*w*_{rev}= â€“2.303*nRT*log_{10}**Work done in irreversible isothermal expansion**

Irreversible isothermal expansions observed are

**(i) free expansion and (ii) intermediate expansion.**Since the external pressure is zero in free expansion, the work done is zero, e.g., expansion of a gas in vacuum.

The external pressure is less than gas pressure in the case of intermediate expansion. Thus, the work done when volume changes from

*V*_{1}to*V*_{2}isSince,

*P*_{ext}is less than the pressure of the gas, the work done during intermediate expansion is numerically less than that during reversible isothermal expansion in which*P*_{ext}is almost equal to*P*.# Adiabatic Expansion

From the first law of thermodynamics, Î”

*E*=*q*+*w*.In an adiabatic expansion,

*q*= 0; therefore, Î”*E*=*w.*The molar heat capacity at constant volume of an ideal gas is given by

or

*dE*=*C*Ã—_{V}*dT*and for finite changes, Î”

*E*=*C*Ã— Î”_{V}*T*=*w*.The value of Î”

*T*depends on the nature of process (i.e., reversible or irreversible).**Reversible adiabatic expansion***TV*Î”^{Î³}^{ â€“ 1}= constant or*PV*= constant^{Î³}**Irreversible adiabatic expansion**