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

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Given: The velocity of sound in air is doubled when temperature changes from $0^\circ \text{C}$ to $a^\circ \text{C}$ .

Find: The value of $a$ .

The speed of sound $v$ is proportional to the square root of absolute temperature $T$ .

v \propto \sqrt{T}

Given $v_2 = 2v_1$ . Therefore,

\sqrt{T_2} = 2\sqrt{T_1}

Squaring both sides,

T_2 = 4T_1

Initial temperature,

T_1 = 0^\circ \text{C} = 273 \, \text{K}

So,

T_2 = 4 \times 273 = 1092 \, \text{K}

Convert back to Celsius:

a = 1092 - 273 = 819^\circ \text{C}

Therefore, the value of $a$ is $819$ .

Given: Sound speed in air becomes double.

Find: The corresponding rise in temperature in degree Celsius.

For a gas, speed of sound depends on temperature as

v = \sqrt{\gamma RT/M}

Since $\gamma$ , $R$ and $M$ remain constant for air, we get

v \propto \sqrt{T}

If the final speed is twice the initial speed,

\frac{v_2}{v_1} = 2 = \sqrt{\frac{T_2}{T_1}}

Squaring,

\frac{T_2}{T_1} = 4

Hence,

T_2 = 4T_1

At $0^\circ \text{C}$ ,

T_1 = 273 \, \text{K}

Thus,

T_2 = 4 \times 273 = 1092 \, \text{K}

Now convert Kelvin to Celsius:

a = 1092 - 273 = 819^\circ \text{C}

Therefore, the required answer is $819$ .

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