Question

The speed of sound in oxygen at S.T.P. will be approximately: Given, \( R = 8.3 \, \text{J K}^{-1} \), \( \gamma = 1.4 \))

• A. \( 310 \, \text{m/s} \)
• B. \( 333 \, \text{m/s} \)
• C. \( 341 \, \text{m/s} \)
• D. \( 325 \, \text{m/s} \)

Answer 1

The Correct Option is A

To determine the speed of sound in oxygen at Standard Temperature and Pressure (S.T.P.), we need to use the formula for the speed of sound in gases:

\(v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}}\)

where:

• \(v\) is the speed of sound,
• \(\gamma\) (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 in kg/mol.

For oxygen, the values are:

• Adiabatic index, \(\gamma = 1.4\),
• Molar mass of oxygen, \(M = 32 \, \text{g/mol} = 0.032 \, \text{kg/mol}\),
• Universal gas constant, \(R = 8.3 \, \text{J K}^{-1}\)

At S.T.P., the temperature \(T\) is generally taken as \(273 \, \text{K}\).

Substituting these values into the formula:

\(v = \sqrt{\frac{1.4 \cdot 8.3 \cdot 273}{0.032}}\)

Calculating inside the square root:

\(v = \sqrt{\frac{3170.44}{0.032}}\)

\(v = \sqrt{99076.25}\)

\(v \approx 314.8 \, \text{m/s}\)

Given the options, the closest value is \(310 \, \text{m/s}\), which can be considered as the correct approximation under the given conditions and assumptions.

Therefore, the correct answer is:

\(310 \, \text{m/s}\)

Answer 2

The speed of sound v in a gas is given by:

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

where:

• \(\gamma\) is the adiabatic index (or ratio of specific heats, \(C_p/C_v\)),
• \(R\) is the universal gas constant,
• \(T\) is the absolute temperature in Kelvin,
• \(M\) is the molar mass of the gas in kg/mol.

We are given:

\[ \gamma = 1.4, \quad R = 8.3 \, \text{J/K mol}, \quad T = 273 \, \text{K}, \quad M = 32 \times 10^{-3} \, \text{kg/mol}. \]

Substitute these values into the formula:

\[ v = \sqrt{\frac{1.4 \times 8.3 \times 273}{32 \times 10^{-3}}}. \]

Calculate the expression inside the square root:

\[ 1.4 \times 8.3 = 11.62, \]

\[ 11.62 \times 273 = 3173.26, \]

\[ \frac{3173.26}{32 \times 10^{-3}} = 99164.375. \]

Now take the square root to find v:

\[ v = \sqrt{99164.375} \approx 315 \, \text{m/s}. \]

The closest answer to this calculated value is: 310 m/s.