Question

Two moles of a gas at temperature t and volume v are heated to twice its volume at constant pressure if cp/cv = gamma then increase in internal engergy of the gas

Answer 1

Given details:

No. of moles of the gas (n) = 2

Cp/Cv = ɣ

Upon heating the gas, its volume V is increased to 2V, at a constant Pressure P.

So, Change in volume is given by ΔV = 2V − V = V.

 Solution:

We know that,  PV = nRT   .......... (A)  

Change in internal energy of the gas ΔU = n Cv ΔT  ....... (B)

PV = nRT  

Differentiating equation (A) ,  

P ΔV =  nR ΔT   ( Here, only the values of V and T are changing, other values are constant)

PV = nR ΔT   (because ΔV = V)  

ΔT = PV / nR        

Now, lets substitute above value in equation (B).

ΔU = 2 *Cv PV / 2* R  

      =  Cv * PV / R  

We know that, R = Cp−Cv

      = Cv * PV / (Cp−Cv)  

     = (1/ (1/Cv) *  PV / (Cp−Cv)  

      = PV/[(Cp−Cv )/Cv ]  

      = PV/(ɣ − 1)

 So, Change in internal energy will be PV/(ɣ − 1)