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

Solution.

$V_1=400cm^3;$

$T_1=27^oC=300K;$

$V_2=500cm^3;$

$\gamma=1.4;$

1) $\dfrac{T_2}{T_1}=(\dfrac{V_1}{V_2})^{\gamma-1};$

$T_2=T_1(\dfrac{V_1}{V_2})^{\gamma-1};$

$T_2=300K(\dfrac{400cm^3}{500cm^3})^{0.4}=274K;$

$P_1V_1^\gamma=P_2V_2^\gamma; \dfrac{P_1}{P_2}=(\dfrac{V_2}{V_1})^\gamma;$

$\dfrac{P_1}{P_2}=(\dfrac{500cm^3}{400cm^3})^{1.4}=1.37;$

2)$P_1V_1=P_2V_2 -$ isothermal process;

$\dfrac{P_1}{P_2}=\dfrac{V_2}{V_1} \implies V_2=\dfrac{P_1}{P_2}V_1;$

$V_2=\dfrac{1}{1.37}\sdot500cm^3=365cm^3;$

Answer:1) $T_2=274K;$

2) $V_2=365cm^3.$