Question

By using the ideal gas equation of state, show that change in the internal energy of the same quantity of an ideal gas in adiabatic process is given by
dU= -NRTlnV

Answer 1

We can write the first law of thermodynamics as

$Q = ΔU +W (1)$

where Q is the heat supplied to the system ΔU is the internal energy of a closed system, W is the work done by the system on its surroundings

We can write for adiabatic process

$Q =0 (2)$

We put (2) in (1)

$0 = ΔU +W (3)$

The ideal gas equation of state for isothermal state as

$W= NRT \ln{V} (4)$

Using (3) and (4) we got

$dU= - NRT \ln{V}$