During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of Cp/Cv for the gas is?

P∝T³
PV = nRT
Since P∝T³, we can write P = kT³ where k is a constant.
Substituting this into the ideal gas law, we get:
kT³V = nRT
kT²V = nR
Since n and R are constants, kT²V is also a constant. Let's call this constant C.
Therefore, T²V = C
For an adiabatic process, we have PVγ = constant, where γ = Cp/Cv
We have T²V = C. From the ideal gas law, we can write V = nRT/P. Substituting this into T²V = C:
T²(nRT/P) = C
T³ / P = C'
where C' is another constant.
Since P∝T³, we have P = kT³
Then T³ / P = 1/k = constant
This means that PV^γ = constant implies P(T³)^γ = constant
Comparing this with T³ / P = constant, we can see that the power of T and P are related as follows: 3/γ = 1/k
Since T³/P = constant, we can write P∝T³
For an adiabatic process, PVγ = constant. We also have PV = nRT, which we can write as V = nRT/P
Substituting V into PVγ = constant, we get P(nRT/P)γ = constant
P^(1-γ)Tγ = constant
Since P∝T³, we can write P = AT³ (where A is a constant)
Substituting this into P^(1-γ)Tγ = constant, we get (AT³)^(1-γ)Tγ = constant
A^(1-γ)T³(1-γ)Tγ = constant
A^(1-γ)T^(3(1-γ)+γ) = constant
For this to be constant, the exponent of T must be 0
3(1-γ) + γ = 0
3 - 3γ + γ = 0
3 - 2γ = 0
2γ = 3
γ = 3/2