The pressure of a mass of air at 20°C is halved adiabatically. If the ratio of C_p to C_v for air is 1.41, calculate the resulting temperature.

Question

Quizplus · Accepted Answer

Since the process is adiabatic, we can use the adiabatic relation:
pV^γ = constant
where p is the pressure, V is the volume, and γ is the ratio of specific heats.
If the pressure is halved, then the final pressure is p/2. Since the volume V is not given, we can use the fact that the initial and final conditions are both on the same adiabat to relate the initial and final volumes:
(p/2)V^γ = pV^γ
Simplifying gives Vf = √2 Vi, where Vi is the initial volume and Vf is the final volume.
Now we can use the first law of thermodynamics, which for an adiabatic process reduces to:
PV^γ = constant
or
T = (P/V)^(1-γ) = constant * P^(γ-1)
where T is the temperature and P is the pressure.
Since the initial and final conditions are both on the same adiabat:
P1/2 * √2 Vi = P Vi
or
Pf = P/√2
Also, since the ratio of specific heats for air is 1.41, we have:
γ = 1.41
Plugging in all the values, we get:
Tf = Ti * (Pf/Pi)^(γ-1)
= 20°C * ((P/√2)/P)^(1.41-1)
= 20°C * (√2)^(-0.41)
= 20°C * 0.802
= 16.04°C
Therefore, the resulting temperature is approximately -33.0°C (Choice C).

The Pressure of a Mass of Air at 20°C Is