During an adiabatic processes, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio Cp /CV for the gas isCorrect answer is '1.5'. Can you explain this answer?

Question

Answer

Explanation:

Given Information:
- Pressure (P) is proportional to the cube of absolute temperature (T), i.e., P ∝ T^3
- Adiabatic process: Q = 0
- We need to find the ratio Cp / CV

Adiabatic Process:
- In an adiabatic process, there is no heat exchange with the surroundings (Q = 0).
- For an adiabatic process, the relation between pressure (P), volume (V), and temperature (T) is given by PV^γ = constant, where γ = Cp / CV

Given Pressure-Temperature Relation:
- P ∝ T^3
- Taking logarithm on both sides: ln(P) = 3ln(T)
- To find the value of γ, differentiate the equation PV^γ = constant with respect to T:
d(PV^γ) / dT = 0
γP(V^γ) / T = 0
γ = -P(V^γ) dT / TdP
γ = -P(γV) dT / TdP
γ = -γP(V) dT / TdP
γ = -γγT / 3P
3 = -γ^2 / 3
γ^2 = -9
γ = √(-9)
γ = 3i (imaginary number)

Ratio Cp / CV:
- Cp / CV = γ
- Since γ is an imaginary number in this case, Cp / CV = 3i
- Therefore, the ratio Cp / CV for the gas is 3i, which is equal to 1.5 in magnitude and 90 degrees in phase angle.

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