Question

An ideal gas at pressure $2.5\times {10}^5$ Pa and temperature 300 K occupies 100 cc. It is adiabatically compresses to half its original volume. Calculate

(a) The final pressure

(b) The final temperature and

(c) The work done by the gas in the process. Take $\gamma =1.5$

Answer 1

$P_1=2.5\times {10}^5$ Pa,

$T_1$ = 300 K

$V_1$ = 100 CC,

(a) $P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$

$\Rightarrow 2.5\times {10}^5\times V^{1.55}={\left(\frac{V}{2}\right)}^{1.5}\times P^2$

$\Rightarrow P_2=7.07\times {10}^5$

= $7.1\times {10}^5$

(b) $T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$

= $T_2\times (50{)}^{1.5-1}$

$\Rightarrow 300\times 10=T_2\times 7.07$

$\Rightarrow T_2=424.32K=424K$

(c) Work done by the gas in the process

W = $\frac{mR}{(\gamma -1)}[T_2-T_1]$

= $\frac{P_1{V}_1}{(\gamma -1)}[T_2-T_1]$

= $\frac{2.5\times 10}{300\times 0.5}\times 124$

= 20.67 = 21 J