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