Question

Air (assume it to be a mixture of $N_2$ and $O_2$) at pressure $P$ and volume $V$ is expanded adiabatically. The volume increases by $5\%$. What will be the correct relation between initial and final pressure$\left(P_1\right)$.

• A. $P_1={\left[\frac{100}{105}\right]}^{\frac{7}{5}}\times P$
• B. $P_1={\left[\frac{100}{105}\right]}^{\frac{5}{3}}\times P$
• C. $P_1={\left[\frac{105}{100}\right]}^{\frac{7}{5}}\times P$
• D. $P_1={\left[\frac{105}{100}\right]}^{\frac{5}{3}}\times P$

Answer 1

The correct option is A $P_1={\left[\frac{100}{105}\right]}^{\frac{7}{5}}\times P$

We know for an adibatic expansion
$P{V}^{\gamma }=\text{constant}........\left(1\right)$
$P_1\times {\left(\frac{105V}{100}\right)}^{\gamma }=\text{constant}.......\left(2\right)$
Now dividing the equation (2) by equation (1), we get
$\frac{P_1}{P}\times {\left(\frac{105}{100}\right)}^{\gamma }=1\Rightarrow \frac{P_1}{P}={\left[\frac{100}{105}\right]}^{\frac{7}{5}}$
($\gamma$ for $N_2$ or $O_2=\frac{7}{5}$)
$P_1={\left[\frac{100}{105}\right]}^{\frac{7}{5}}\times P$