CBSE Class 11-science Answered

why the work done in isothermal reversible expansion is more than that of irreversible expansion

Asked by Ankana Paul | 01 Nov, 2010, 09:45: PM

Expert Answer

(i) Work done in reversible isothermal expansion: Consider an ideal gas enclosed in a cylinder fitted with a weightless and frictionless piston. The cylinder is not insulated. The external pressure, P_ext is equal to pressure of the gas, P_gas.

P_gas=P_gas= P

If the external pressure is decreased by an infinitesimal amount dP, the gas will expand by an infinitesimal volume, dV. As a result of expansion, the pressure of the gas within the cylinder falls to P_gas – dP, i.e., it becomes again equal to the external pressure and, thus, the piston comes to rest. Such a process is repeated for a number of times, i.e., in each step the gas expands by a volume dV.

Since the system is in thermal equilibrium with the surroundings, the infinitesimally small cooling produced due to expansion is balanced by the absorption of heat from the surroundings and the temperature remains constant throughout the expansion.

The work done by the gas in each step of expansion can be given as,

d_w = –(P_ext– dP) dV = –P_ext.dV – dP.dV

The product of two infinitesimal quantities, is negligible.

The total amount of work done by the isothermal reversible expansion of the ideal gas from volume V₁ to volume V₂ is, given as,

w = –nRT loge V₂/V₁ or w = –2.303nRT log₁₀ V₂/V₁

At constant temperature, according to Boyle’s law,

P₁V₁ = P₂V₂ or V₂/V₁ = P₁/P₂ So, w = –2.303nRTlog₁₀ P₁/P₂

(ii) Work done in irreversible isothermal expansion : Two types of irreversible isothermal expansions are observed, i.e., (a) Free expansion and (b) Intermediate expansion. In free expansion, the external pressure is zero, i.e., work done is zero when gas expands in vacuum. In intermediate expansion, the external pressure is less than gas pressure. So, the work done when volume changes from V₁ to V₂ is given by

Since P_ext is less than the pressure of the gas,

the work done during intermediate expansion is numerically less than the work done during reversible isothermal expansion in which P_ext is almost equal to P_ext.