Isothermal Expansion : In the isothermal expansion, ΔT = 0; ΔE = 0.

         According to first law of thermodynamics,

                        ΔE = q+w        thus q = -w

         This shows that in the isothermal expansion, the work is done by the system at the expense of the heat absorbed.

         Since for isothermal process, ΔE and ΔT are zero correspondingly, therefore, ΔH = 0

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

      P_ext = P_gas = P

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

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

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

         dP.dV the product of the two infinitesimal amount, can be neglected.

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

         At constant temperature, according to the Boyle's law,

         Isothermal compression work of the ideal gas might be derived similarly and it has exactly same value with positive sign.

         (ii) Work done in irreversible isothermal expansion : The two types of irreversible isothermal expansions are observed, which are stated as follows
(a) Free expansion
(b) Intermediate expansion.
In free expansion, external pressure is zero, which means work done is zero when the gas expands in vacuum. In the intermediate expansion, external pressure is less than the pressure of gas. So, the work done when the volume changes from V₁ to V₂ is can be given as

         Since P_ext is less than the pressure of the gas, the work done during the intermediate expansion is numerically less than work done during this reversible isothermal expansion in which the P_ext is nearly equal to P_gas.

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