Question 77: Waves

Question

The velocity of sound in air is doubled when the temperature is raised from $0^{\circ} \mathrm{C}$ to $\alpha{ }^{\circ} \mathrm{C}$. The value of $\alpha$ is $\_\_\_\_$ .

Accepted Answer

The velocity of sound in an ideal gas depends on the absolute temperature of the medium.
The speed of sound (v) in air is given by the formula :
$$ \mathrm{v}=\sqrt{\frac{\gamma \mathrm{RT}}{\mathrm{M}}} $$
From this, we can see that :
$$ \mathrm{v} \propto \sqrt{\mathrm{~T}} $$
where T is the absolute temperature in Kelvin $(\mathrm{K})$.
Initial State :
Temperature $=\mathrm{T}_1=0^{\circ} \mathrm{C}=(0+273) \mathrm{K}=273 \mathrm{~K}$
Velocity $=\mathrm{v}_1$
Final State :
Temperature $=\mathrm{T}_2=\alpha^{\circ} \mathrm{C}=(\alpha+273) \mathrm{K}$
Velocity $=\mathrm{v}_2=2 \mathrm{v}_1($ since the velocity is doubled $)$.
Using the proportionality $\mathrm{v} \propto \sqrt{\mathrm{T}}$ :
$$ \frac{\mathrm{v}_2}{\mathrm{v}_1}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}} $$
Substituting the values :
$$ \frac{2 v_1}{v_1}=\sqrt{\frac{\alpha+273}{273}} $$
$$\Rightarrow $$ $4=\frac{\alpha+273}{273}$
$$\Rightarrow $$ $4 	imes 273=\alpha+273$
$$\Rightarrow $$ $\alpha=1092-273$
$$\Rightarrow $$ $$ \alpha=819 $$
Therefore, the value of $\alpha$ is 819 . Hence, the correct answer is $\mathbf{8 1 9}$.

Waves

The velocity of sound in air is doubled when the temperature is raised from 0^{\circ} \mathrm{C} to \alpha{ }^{\circ} \mathrm{C}. The value of \alpha is \_\_\_\_ .

Choose the correct answer: