Question

The Volume of air increases by $5$% in its Adiabatic Expansion. The percentage decrease in its pressure will be

• A. $6\%$
• B. $7\%$
• C. $8\%$
• D. $9\%$

Answer 1

The correct option is B

$7\%$

The Explanation for Correct option:

(b) $7\%$

• Equation of States:

$$\begin{gathered} d\mathrm{Q}=0 \\ {\mathrm{\eta C}}_{\mathrm{v}}d\mathrm{T}+\mathrm{P}d\mathrm{V}=0 \end{gathered}$$

• On solving:

$$\begin{gathered} \mathrm{\gamma }\frac{\mathrm{dV}}{\mathrm{V}}+\frac{\mathrm{dP}}{\mathrm{P}}=0 \\ {\mathrm{PV}}^{\mathrm{\gamma }}=\mathrm{constant} \end{gathered}$$

• On Differentiating:

$$\begin{gathered} \mathrm{dP}.{\mathrm{V}}^{\mathrm{\gamma }}+{\mathrm{\gamma V}}^{\mathrm{\gamma }-1}.\mathrm{PdV}=0 \\ \mathrm{dP}=-\mathrm{\gamma }\frac{\mathrm{P}}{\mathrm{\nu }}.\mathrm{dV} \\ \frac{\mathrm{dP}}{\mathrm{P}}=-\mathrm{\gamma }\frac{\mathrm{dV}}{\mathrm{V}} \\ \frac{\mathrm{dP}}{\mathrm{P}}=-(1.4)\times 5\% \\ \frac{\mathrm{dP}}{\mathrm{P}}=-7\% \end{gathered}$$

The Explanation for the Incorrect options:

Since, the correct percentage decrease is $7\%$. Therefore, options (a), (c) and (d) stand incorrect.

Hence, The Correct option is (b) $7\%$