Question

How does the speed of sound in a gas change when you raise the temperature from$\mathbf{0}\mathbf{°}\mathit{C}\mathbf{}\mathit{T}\mathit{O}\mathbf{}\mathbf{20}\mathbf{°}\mathit{C}$ ? Explain briefly.

Short answer

The speed of sound increases as the temperature of the gas increases and vice-versa is also true.

Step-by-step solution

Understanding the sound speed

In this problem, the concept of sound speed will be used to determine the dependency of sound speed on a variable like a temperature, etc. For the movement of a sound wave, the requirement of a specific medium is compulsory.

Evaluating how the speed of sound in a gas changes when you raise temperature from 0°C to 20°C.

The speed of sound is the distance travelled per unit of time by a sound wave as

it propagates through an elastic medium.

The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on the frequency and pressure in ordinary air, deviating slightly from ideal behaviour.

The mathematical expression to evaluate the sound speed at different temperature is expressed as follows:

$v=\sqrt{\gamma RT}$

Here, $V$ represents the speed of sound, $\gamma$ represents the ratio of specific heats of medium (ideal gas), $R$ represents the gas constant and $T$ represents the absolute temperature of medium (ideal gas).

Thus, it is reasonable that the speed of sound in a gas depend on the square root of temperature. For air, atrole="math" localid="1655487640043" $\mathit{0}\mathit{°}C$ the speed of sound is 331 m/s, whereas at $\mathit{20}\mathit{°}C$it is 343 m/s, less than an increment of 4%.

Thus, the speed of sound varies in a direct proportion of the square root of absolute temperature of the gas.