Explanation

For an ideal gas, the velocity of sound is given by:

$$ v = \sqrt{\frac{\gamma RT}{M}} $$

Hence, the velocity of sound is proportional to the square root of absolute temperature:

$$ v \propto \sqrt{T} $$

Initial temperature:

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

Let the initial velocity be $v_1$.

Final temperature:

$$ T_2 = \alpha^\circ\text{C} = (\alpha + 273)\ \text{K} $$

Given that the velocity is doubled:

$$ v_2 = 2v_1 $$

Using the proportionality relation:

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

Substituting values:

$$ \frac{2v_1}{v_1} = \sqrt{\frac{\alpha + 273}{273}} $$

$$ 2 = \sqrt{\frac{\alpha + 273}{273}} $$

Squaring both sides:

$$ 4 = \frac{\alpha + 273}{273} $$

$$ 4 \times 273 = \alpha + 273 $$

$$ \alpha = 1092 - 273 $$

$$ \alpha = 819 $$

Final Answer: $$ \boxed{819} $$

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