The speed of sound in oxygen at S.T.P. will be approximately: Given, \( R = 8.3 \, \text{J K}^{-1} \), \( \gamma = 1.4 \))
- \( 310 \, \text{m/s} \)
- \( 333 \, \text{m/s} \)
- \( 341 \, \text{m/s} \)
- \( 325 \, \text{m/s} \)
The Correct Option is A
Approach Solution - 1
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:
\(310 \, \text{m/s}\)
Approach Solution -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.
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