Question

A mass of ideal gas of volume 400 cm3 at a temperature of 27 ˚C expands adiabatically until its volume is 500 cm3. Calculate the new temperature. The gas is then compressed isothermally until its pressure returns to the original value. Calculate the final volume of the gas. Assume γ = 1.40.

Answer 1

For the adiabatic process, we have

$$T_1V_1^{\gamma-1}=T_2V_2^{\gamma-1},\\\space\\ T_2=T_1\bigg(\frac{V_1}{V_2}\bigg)^{{\gamma-1}},\\\space\\ T_2=(27+273)\bigg(\frac{400}{500}\bigg)^{\gamma-1}=274\text{ K, or 1°C.}$$

Now, calculate the initial pressure:

$$P_1V_1^\gamma=P_2V_2^\gamma,\\ P_1^{\gamma-1}T_1^\gamma=P_2^{\gamma-1}T_2^\gamma,\\\space\\ P_1=\\ P_2=$$

According to the ideal gas law, for the isothermal compression we have

$$P_2V_2=P_1V_3.$$

Also, according to the same law, for the second and initial states of gas we have

$$\frac{P_2V_2}{T_2}=\frac{P_1V_1}{T_1}.$$

Combine the last two equations to determine $V_3$:

$$V_3=V_1\frac{T_2}{T_1}=400\cdot\frac{274}{300}=364\text{ cm}^3.$$