91Ó°ÊÓ

1. The pressure is $p=1.00\text{atm}=1.01×{10}^5\text{Pa}$.
2. The speed of sound is $v_s=331\text{m}/\text{s}$.
3. The air at temperature$0.000°\text{C}$ .

The density is $\mathrm{ÒÏ}=1.29×{10}^{-3}\text{g}/{\text{cm}}^3=1.29\text{kg}/{\text{m}}^3$.

By using the formula, from problems19-91,for the speed of the sound in gas, find the ratio of the molar-specific heat at constant pressure to that at constant volume, The speed with which the sound travels in gas is given as,

${\mathbf{v}}_{\mathbf{s}}\mathbf{=}\sqrt{\frac{\mathbf{γ\pm è}}{\mathbf{ÒÏ}}}$

where, ${\mathbf{v}}_{\mathbf{s}}$is velocity,$\mathrm{ÒÏ}$is the density, and p is pressure.

From the problems19-91, The speed with which the sound travels in gas is given as,

${\text{v}}_s=\sqrt{\frac{\mathrm{γ}\text{p}}{\mathrm{ÒÏ}}}$

Where $\mathrm{γ}={\text{C}}_p/{\text{C}}_V$

Squaring both sides,

${\text{v}}_s^2=\frac{\mathrm{γ}\text{p}}{\mathrm{ÒÏ}}$

Therefore, $\mathrm{λ}$ is given by,

$\mathrm{γ}=\frac{\mathrm{ÒÏ}{v}_s^2}{p}$

$$\begin{gathered} \mathrm{γ}=\frac{1.29\text{kg}/{\text{m}}^3×{\left(331\text{m}/\text{s}\right)}^2}{1.01×{10}^5\text{Pa}} \\ \mathrm{γ}=1.40 \end{gathered}$$

Hence, the ratio of the molar-specific heat at constant pressure to that at constant volume, of air is $\mathrm{γ}=1.40$.