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A sound wave passing through an ideal gas at NTP produces a pressure change of 0.001 dyne $/ \mathrm{cm}^2$ during adiabatic compression. The corresponding change in temperature $(\gamma=1.5$ for the gas and atmospheric pressure is $1.013 \times 10^6$ dyne $/ \mathrm{cm}^2$ ) is
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The correct answer is:
$8.97 \times 10^{-8} \mathrm{~K}$
$T^\gamma p^{1-\gamma}=$ constant
or
$T^\gamma=p^{\gamma-1}$
$\begin{aligned} T & =p\left(\frac{\gamma-1}{\gamma}\right) \\ \therefore \quad \frac{\Delta T}{T} & =\frac{\gamma-1}{\gamma} \times \frac{\Delta p}{p} \\ \frac{\Delta T}{T} & =\left(\frac{1.5-1}{1.5}\right) \times \frac{0.001}{1.013 \times 10^6} \\ \Delta T & =8.98 \times 10^{-8} \mathrm{~K}\end{aligned}$
or
$T^\gamma=p^{\gamma-1}$
$\begin{aligned} T & =p\left(\frac{\gamma-1}{\gamma}\right) \\ \therefore \quad \frac{\Delta T}{T} & =\frac{\gamma-1}{\gamma} \times \frac{\Delta p}{p} \\ \frac{\Delta T}{T} & =\left(\frac{1.5-1}{1.5}\right) \times \frac{0.001}{1.013 \times 10^6} \\ \Delta T & =8.98 \times 10^{-8} \mathrm{~K}\end{aligned}$
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