The speed of sound in oxygen `(O_2)` at a certain temperature is `460ms^-1`. The speed of sound in helium (He) at the same temperature will be (assume both gases to be ideal)

Options

A. `460 ms^(-1)`
B. `500 ms^(-1)`
C. `650 ms^(-2)`
D. `330 ms^(-1)`

Answer

To find the speed of sound in helium (He) given the speed of sound in oxygen (O₂) at the same temperature, we can use the formula for the speed of sound in a gas:

\[ v = \sqrt{\frac{\gamma RT}{M}} \]

where:
- \(v\) is the speed of sound,
- \(\gamma\) is the adiabatic index (ratio of specific heats),
- \(R\) is the universal gas constant,
- \(T\) is the absolute temperature,
- \(M\) is the molar mass of the gas.

### Step 1: Write the ratio of speeds of sound in helium and oxygen

Since the temperature \(T\) and the gas constant \(R\) are the same for both gases, we can write the ratio of the speeds of sound in helium and oxygen as:

\[ \frac{v_{He}}{v_{O_2}} = \sqrt{\frac{\gamma_{He}}{\gamma_{O_2}} \cdot \frac{M_{O_2}}{M_{He}}} \]

### Step 2: Determine the values of \(\gamma\) for helium and oxygen

- For helium (He), which is a monatomic gas, the degrees of freedom \(f\) is 3. Thus, we can calculate \(\gamma\) as:

\[ \gamma_{He} = 1 + \frac{2}{f} = 1 + \frac{2}{3} = \frac{5}{3} \]

- For oxygen (O₂), which is a diatomic gas, the degrees of freedom \(f\) is 5. Thus, we calculate \(\gamma\) as:

\[ \gamma_{O_2} = 1 + \frac{2}{f} = 1 + \frac{2}{5} = \frac{7}{5} \]

### Step 3: Determine the molar masses of helium and oxygen

- The molar mass of helium \(M_{He}\) is 4 g/mol.
- The molar mass of oxygen \(M_{O_2}\) is 32 g/mol.

### Step 4: Substitute values into the ratio

Now we can substitute the values of \(\gamma\) and \(M\) into the ratio:

\[ \frac{v_{He}}{v_{O_2}} = \sqrt{\frac{\frac{5}{3}}{\frac{7}{5}} \cdot \frac{32}{4}} \]

### Step 5: Simplify the expression

Calculating the ratio inside the square root:

\[ \frac{5}{3} \div \frac{7}{5} = \frac{5}{3} \cdot \frac{5}{7} = \frac{25}{21} \]

Now, substituting the molar mass ratio:

\[ \frac{32}{4} = 8 \]

Thus, we have:

\[ \frac{v_{He}}{v_{O_2}} = \sqrt{\frac{25}{21} \cdot 8} = \sqrt{\frac{200}{21}} \approx \sqrt{9.52} \]

### Step 6: Calculate \(v_{He}\)

Given that \(v_{O_2} = 460 \, \text{m/s}\):

\[ v_{He} = v_{O_2} \cdot \sqrt{9.52} \approx 460 \cdot 3.08 \approx 1414.8 \, \text{m/s} \]

### Final Answer

The speed of sound in helium at the same temperature is approximately:

\[ \boxed{1420 \, \text{m/s}} \]