Question

A sound wave passing through the air at NTP $0^{\circ} C$ and $1$ atm produces a pressure of $0.001$ dyne $cm ^{-2}$ during a compression. The corresponding change in temperature (given $\gamma=1.5$ and assume gas to be ideal) is

• A. $8.97\times 10^{- 4}K$
• B. $9.87\times 10^{- 6} \, K$
• C. $8.97\times 10^{- 8} \, K$
• D. $9.87\times 10^{- 4} \, K$

Answer 1

Answer: (C)

$T ^{\gamma} P ^{1-\gamma}=$ constant
Differentiating
$\gamma T ^{\gamma-1} dT P^{1-\gamma}+ T ^{\gamma} 1-\gamma P ^{-\gamma} dP =0$
or, $d T =\frac{{\gamma -1}^ T }{\gamma^{ P }} dP$
or, $dT =\frac{1.5-1}{1.5} \frac{273}{76 \times 13.6 \times 981} \times 0.001$
$\Rightarrow 8.97 \times 10^{-8} K$