The volume of one mode of an ideal gas with adiabatic exponent `gamma` is varied according to the law `V = a//T`, where a is constant . Find the amount of heat obtained by the gas in this process, if the temperature is increased by `Delta T`.

Explanation

`Delta W = int_(T)^(T + Delta T) ``_(p)dV`, `_(p)V = RT` (always) and here `V = a//T`
`:. Delta W = int_(T)^(T + Delta T) (RT^(2))/(a) (-(a)/(T^(2)) dT) = - int_(T)^(T + Delta T) RdT - R Delta T`
`Delta U = int_(T)^(T + Delta T) C_(V) dT = (R )/(gamma - 1) Delta T` `( :' C_(V) (R)/(gamma - 1))`
`:. Delta Q = Delta U + Delta W = (R )/(gamma - 1) Delta T - R Delta T = (2 - gamma)/(gamma - 1) R Delta T`