Animated Solution for Physics - Thermodynamics: The speed of sound in oxygen (${\text{O}}_2$) at a certain temperature is $460{\text{ ms}}^{-1}$. The speed of sound in helium (He) at the same temperature will be (assume both gases to be ideal)

Oxygen vs. Helium

Imagine two identical tubes, one filled with Oxygen gas (${\text{O}}_2$) and the other with Helium gas ($\text{He}$). Both gases are maintained at the exact same temperature $T$. We are given that the speed of sound in the Oxygen tube is $460\text{ m/s}$. Our mission is to find out how fast sound travels through the Helium tube.

To do this, we need to understand what governs the speed of sound in a gas.

The speed of sound $v$ in an ideal gas is given by the beautiful Laplace formula:

Here, $\gamma$ is the adiabatic index (ratio of specific heats), $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass of the gas. Since $R$ and $T$ are constant for both gases, the speed of sound will be entirely dictated by the interplay between $\gamma$ and $M$.

Let's first look at Oxygen. Oxygen is a diatomic gas, meaning its molecules consist of two atoms bonded together. Because of this structure, it has both translational and rotational degrees of freedom, giving it an adiabatic index of ${\gamma }_{{\text{O}}_2}=\frac{7}{5}$.

Its molar mass is $M_{{\text{O}}_2}=32\text{ g/mol}$. Plugging these into our master equation, we get:

Now, let's shift our focus to Helium. Helium is a noble gas, which means it is monatomic. Its atoms fly around individually, possessing only translational degrees of freedom. This gives Helium an adiabatic index of ${\gamma }_{\text{He}}=\frac{5}{3}$.

Its molar mass is much lighter, just $M_{\text{He}}=4\text{ g/mol}$. Substituting these values, the speed of sound in Helium is:

Since we don't know the exact temperature $T$, we can cleverly eliminate it by taking the ratio of the two speeds. Let's divide $v_{\text{He}}$ by $v_{{\text{O}}_2}$:

The $RT$ terms cancel out perfectly, leaving us with a ratio of pure numbers:

Let's carefully simplify this massive fraction inside the square root. Rearranging the terms, we bring the denominators of the lower fraction to the top:

Simplifying the numbers, $4$ goes into $32$ exactly $8$ times. So the numerator becomes $5\times 8\times 5=200$. The denominator is $3\times 7=21$.

Now, we multiply this ratio by the given speed of sound in Oxygen ($460\text{ m/s}$):

Since $\frac{200}{21}\approx 9.52$, its square root is approximately $3.086$.

Notice how much faster sound travels in Helium ($1420\text{ m/s}$) compared to Oxygen ($460\text{ m/s}$)! This is primarily because Helium is much lighter than Oxygen. According to the formula, $v\propto \frac{1}{\sqrt{M}}$. The lighter the gas particles, the faster they can accelerate and transmit the kinetic energy of the sound wave. This is exactly why inhaling Helium makes your voice sound so high-pitched!