What is an isothermal process? State essential conditions for such a process to take place. Show analytically that work done by one mole of an ideal gas during isothermal expansion from volume `V_1` to volume `V_2` is given by `= "RT log"_(e) V_2/V_1` . What is the change in internal energy of a gas, which is compressed isothermally?

Explanation

A process in which, temperature of gas remains constant is said to be an isothermal process.
In this Process. Heat exchanged is used entirely as work.
In an isothermal process, dQ = dW and dU = 0, dT = 0 and PV = constant
To take a gas through an isothermal process, it must be placed in a container fitted with a piston and the container should have conducting walls. The process is carried out by varying the pressure of the gas very slowly by adjusting the mass kept on the piston. The slow speed of the process allows the gas to always remain in thermal equilibrium with its surroundings. As the surroundings are at a constant temperature, the gas temperature also remains constant. Workdone in Isothermal Process `W = int p.dV = int(nRT)/V .dV" By PV = nRT"`
`= underset(V_1)overset(V_2)(int)"nRT" (dV)/V = "nRT"|l n V|_(V_(1))^(V_(2))`
`W = "nRT ln"V_2/V_1`
For n = 1
`"W = RT log" V_2/V_1`
Change in internal energy, `Delta U = nC_V Delta T` for any process. So, for isothermal compression `Delta U = 0` aa `Delta T =0`