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