Question

The velocity of sound in air is doubled when the temperature is raised from $0 ^\circ$C to $a ^\circ$C. The value of a is ___.

Hint:

Always use Kelvin scale for temperature in gas laws and wave velocity formulas ($v = \sqrt{\gamma RT/M}$).

Answer 1

Correct Answer: 819

To solve for the temperature \(a^\circ\)C at which the velocity of sound in air doubles, we start with the formulas for the velocity of sound in air at different temperatures. The velocity of sound \(v\) in air at temperature \(T\) in Celsius is given by \(v = v_0 \sqrt{1 + \frac{T}{273}}\), where \(v_0\) is the velocity of sound at \(0^\circ\)C. Since we are doubling the velocity, we have: 1. At \(0^\circ\)C, \(v_1 = v_0\). 2. At \(a^\circ\)C, \(v_2 = 2v_0 = v_0\sqrt{1 + \frac{a}{273}}\). Equating for \(v_2 = 2v_1\), we have: \[2v_0 = v_0\sqrt{1 + \frac{a}{273}}\] 3. Dividing both sides by \(v_0\), we get: \[2 = \sqrt{1 + \frac{a}{273}}\] 4. Squaring both sides, we obtain: \[4 = 1 + \frac{a}{273}\] 5. Simplifying, we find: \[\frac{a}{273} = 3\] 6. Solving for \(a\), we have: \[a = 3 \times 273 = 819\] Thus, the computed value for \(a\) is 819°C. This falls within the expected range of 819 to 819, validating our solution.