dU =nCV dT for n moles of gas.

I am able to define the molar heat capacity but do not know how to show that dU =nCV dT . I always thought that this was just an experimental result. Any suggestions?

(b). State Carnot’s Theorem. Using a schematic diagram and reductio ad absurdum

(proof by contradiction), show that the Theorem must be true or the 2

I have no idea how to go about this part of the question so any help would be welcome.

(c). What is the mean free path for a hypothetical molecule of radius 0.15 nm at 300K and 10^5 Pa? How does this compare to the average molecular separation under this condition?

I was able to do to find the mean free path but I do not know what the average molecular seperation is or how to calculate it.

I can answer the first one.
From second law-
dQ = dU +pdV. Since we are talking of the specific heat at constant volume, dV=0 and hence pdV = 0. The above equation reduces to-
dQ = dU .........................(1)

Now heat dQ required to raise the temperature of n mole of gas at constant volume by dT is given by-
dQ =nCvdT .......................(2)
From equation (1) and (2) we have
dU = nCvdT

So (1) is for constant volume and case (2) is for a changing volume? Is this formula true in all cases of ideal gasses, i.e. in non-adiabatic and non- isothermal processes?

1 dq=du+dw is a fundamental law of nature. its the first law of thermodynamics

at constant volume dw=0
hence dq=du
hence dq=nCvdT
dq/dT=nCV
clearly dq/dT=nCV only for constant volume processes
for other quasistatic processes involving ideal gases,
dq=nCdT where C is the molar heat capacity of THAT PROCESS
at constant volume C=Cv
at constant pressure C=Cp
for adiabatic processesC=0 as dq=0
now for isothermal processes, C is not defined as theres no RISE IN TEMPERATURE since all the heat supplied is utilized by the system for doing work..