Let me first start with the meaning of an Isothermal process. It is a change in system in which the temperature remains constant. The term Iso means equaland thermal means heat. This occurs when a normal system comes in contact with a heat engine. This process is a very slow process where a normal system continuosly adjusts to the temperature of the heat engine until there is no transfer of heat between both the systems or until thermal equilibrium is maintained.

If a gas is kept at an isothermal condition, the product of pv is a constant. This value of this constant is nRT.

pv = nRT

where,

p = pressure

v = volume

n = number ofmoles of gas present

R = ideal gas constant

T = temperature (constant in this case)

So, pv = constant (depends upon the number of moles of gas present in the system)

Hence p is indirectly proportional to v.

Work is defined as the work done on the system by its surroundings. If the system is compressed then the work done is positive and if the system expands then the work done will be negative. In our case the system will expand by absorbing the temperature from the heat engine. System will expand means the volume of the system will expand. The formula for work done is:

=- pdV

       = -(nRT)dV/V

       = -nRTln(V)

       = -nRT ln ()

       = -nRT ln ()

       = nRT ln ()