Question

A diatomic ideal gas is heated at constant volume until its pressure is doubled. It is again heated at constant pressure until its volume is doubled. The molar heat capacity for the whole process is kR. Find the value of k.

Answer 1

Diatomic ideal gas =>  γ = 1.4
C_v = 5/2 R
C_p = 7/2 R

     C_p = γ C_v

Let us say the mass of the gas in the context to be n moles.   Let the molar mass of the gas be  M.

   P V / T = constant for an ideal gas.

P1, V1, T1      ====>  heated  ===>  P2, V1,  T2
        P2 = 2 P1    =>    T2= 2 T1    as volume is constant.        ---- (1)

2 P1,  V1,  2 T1    ===> heated  ===>  2P1, V3, T3
       V3 = 2 V1    =>   T3 = 2 * (2 T1) = 4 T1      as pressure is constant    --- (2)


During the 1st constant volume heating process:
             ΔQ1  =  n C_v  ΔT        ,  W = 0 as V is constant
                   = n C_v  (2T1 - T1) = n C_v * T1

During the 2nd constant pressure heating process
           ΔQ2  =  n C_p ΔT  =  n C_p * (4T1 - 2T1) = 2 n C_p  T1
                     = 2 n γ C_v T1

Total heat absorbed by the system :
    ΔQ1 + ΔQ2 =     n C_v T1 + 2 n γ C_v T1  =  n T1 (1 + 2 γ) C_v

The total change in the temperature of the system:
       T3 - T1 = ΔT = 4T1 - T1 = 3 T1
      
 Molar Heat capacity of the system
   =   C_m  = Total heat energy supplied / number of moles
   => C_m = [ ΔQ1 + ΔQ2 ] /  [ n ΔT ]
   => C_m =  [ n T1 (1 + 2 γ) C_v  ]  / [ n 3 T1 ]
               =  (1 + 2 γ) C_v / 3

   C_m  = (1 + 2 * 7/5)  5/2 R  / 3
           =  19 / 6 * R

=>  k = 19/6