Question

The volume of an ideal gas (γ = 1.5) is changed adiabatically from 4.00 litres to 3.00 litres. Find the ratio of (a) the final pressure to the initial pressure and (b) the final temperature to the initial temperature.

Answer 1

(a) The ratio of the final pressure to the initial pressure is $1.54$

(b) The ratio of the final temperature to the initial temperature is $1.154$

Explanation:

Given data,

The ideal gas volume is  

$\gamma=1.5$

Because that cycle is adiabatic, $P V^{\gamma}=\text { constant }$

(a)

$P_{1} V_{1}^{\gamma}=P_{2} V_{2}^{\gamma}$

Given, $V_{1}=4 L$

           $V_{2}=3 L$

We do need to find out $\frac{P_{2}}{P_{1}}$ .

$\frac{P_{2}}{P_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{\gamma}$

$=\left(\frac{4}{3}\right)^{1.5}$

$=1.5396$

$=1.54$

(b) To an adiabatic method, too,

$T V^{\gamma-1}=\text { constant }$

$T_{1} V_{1}^{\gamma-1}=T_{2} V_{2}^{\gamma-1}$

$\frac{T_{2}}{T_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{\gamma-1}$

$=\left(\frac{4}{3}\right)^{0.5}$

$=1.154$