Question 185: Thermodynamics & KTG

Question

Consider the sound wave travelling in ideal gases of $\mathrm{He}, \mathrm{CH}_4$, and $\mathrm{CO}_2$. All the gases have the same ratio $\frac{P}{\rho}$, where $P$ is the pressure and $\rho$ is the density. The ratio of the speed of sound through the gases $\mathrm{V}_{\mathrm{He}}: \mathrm{V}_{\mathrm{CH}_4}: \mathrm{V}_{\mathrm{CO}_2}$ is given by

Accepted Answer

The speed of sound in an ideal gas is given by
$$ v = \sqrt{\gamma \frac{P}{\rho}}, $$
where:
$$\gamma$$ is the ratio of specific heats,
$$P$$ is the pressure, and
$$\rho$$ is the density.
Since the problem states that $$\frac{P}{\rho}$$ is the same for all the gases, the speed of sound in each gas is determined solely by the factor $$\sqrt{\gamma}$$.
Let's determine the appropriate $$\gamma$$ for each gas:
Helium (He):
Helium is a monatomic gas.
For a monatomic gas, $$\gamma = \frac{5}{3}$$.
Therefore, $$v_{\mathrm{He}} \propto \sqrt{\frac{5}{3}}$$.
Methane (CH$_4$):
Methane is a polyatomic gas (a tetrahedral molecule with three rotational degrees of freedom).
For nonlinear polyatomic gases, $$\gamma$$ is typically taken as $$\frac{4}{3}$$.
Thus, $$v_{\mathrm{CH_4}} \propto \sqrt{\frac{4}{3}}$$.
Carbon Dioxide (CO$_2$):
Carbon dioxide is a linear molecule.
For linear molecules, $$\gamma = \frac{7}{5}$$.
Hence, $$v_{\mathrm{CO_2}} \propto \sqrt{\frac{7}{5}}$$.
Therefore, the ratio of the speeds of sound in the gases is:
$$ v_{\mathrm{He}} : v_{\mathrm{CH_4}} : v_{\mathrm{CO_2}} = \sqrt{\frac{5}{3}} : \sqrt{\frac{4}{3}} : \sqrt{\frac{7}{5}}. $$
Comparing this expression with the given options, we see that it matches
Option C
.

Answer

$\sqrt{\frac{7}{5}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}$

Answer

$\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$

Answer

$\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{7}{5}}$

Answer

$\sqrt{\frac{4}{3}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{7}{5}}$

Thermodynamics & KTG

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