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Chapter 14: Problem 9

If you double the pressure of a gas while keeping its density the same, what happens to the sound speed?

Short Answer

Expert verified
When the pressure of a gas is doubled while its density is kept constant, the speed of sound increases by a factor of \(\sqrt{2}\).

Step by step solution

01

Understand the equation for the speed of sound in a gas

The equation for the speed of sound \(v\) in an ideal gas is given by \(v = \sqrt{\frac{P}{\rho}}\), where \(P\) represents the pressure of the gas and \(\rho\) the density of the gas. This equation guides understanding of how the pressure and the density of a gas affect the sound speed in the gas.
02

Identify the given elements

According to the exercise, the pressure of the gas is doubled and the density remains the same. Hence, the new pressure \(P'\) is \(2P\), and the new density \(\rho'\) remains as \(\rho\).
03

Calculate the new speed of sound

Substitute \(P'\) and \(\rho'\) into the sound speed's formula: \(v' = \sqrt{\frac{P'}{\rho'}} = \sqrt{\frac{2P}{\rho}}\). By comparing it with the original speed \(v = \sqrt{\frac{P}{\rho}}\), it can be inferred that the new speed of sound is \(\sqrt{2}\) times the original speed.

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