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The velocity of sound in a gas at pressure P and density d is
A) \[v = \sqrt {\dfrac{{\gamma P}}{d}} \]
B) \[v = \sqrt {\dfrac{P}{{\gamma d}}} \]
C) \[v = \gamma \sqrt {\dfrac{P}{d}} \]
D) \[v = \sqrt {\dfrac{{2P}}{d}} \]

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Last updated date: 20th Jun 2024
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Answer
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Hint: We know that velocity of some in gases is proportional to square root of specific heat ratio, ideal gas constant and absolute temperature and inversely proportional to the square root of its mass. Now, using the ideal gas equation try to find the expression in terms of pressure and density.

Formula Used:
Dependence of velocity of sound on the specific heat ratio, ideal gas constant, absolute temperature and mass of the gas is given by:
\[v = \sqrt {\dfrac{{\gamma RT}}{m}} \] (1)
Where,
v is the velocity of sound in that gas,
\[\gamma \]is specific heat ratio,
R is ideal gas constant,
T is absolute temperature of the gas,
m is the molar mass of the gas.

Ideal gas equation:
$PV = nRT$ (2)
Where,
P is pressure exerted by the gas,
V is the volume of the gas,
n is number of moles.

Complete step by step answer:
Given:
1. Pressure of the gas is P.
2. Density of the gas is $\dfrac{M}{V} = d$, where M is total mass.

To find: Expression for the velocity of sound.

Step 1
First using eq.(2) find an expression for RT as:
$
  PV = nRT \\
  \therefore RT = \dfrac{{PV}}{n} \\
 $ (3)

Step 2
Now, substitute this relation from eq.(3) in eq.(1) to get the final expression for v as:
\[
  v = \sqrt {\dfrac{{\gamma RT}}{m}} \\
   \Rightarrow v = \sqrt {\dfrac{{\gamma \tfrac{{PV}}{n}}}{{\tfrac{M}{n}}}} \\
  \therefore v = \sqrt {\dfrac{{\gamma P}}{{\tfrac{M}{V}}}} = \sqrt {\dfrac{{\gamma P}}{d}} \\
 \]

Correct answer:
Velocity of sound is given by: (a) \[v = \sqrt {\dfrac{{\gamma P}}{d}} \].

Note: This expression of the velocity of sound is known as Laplace’s equation for the velocity of sound in gases. Now, assuming the relationship of eq.(1) we can derive this equation, again assuming laplace’s equation to be true the relation of eq.(1) can be proved. These are actually the same equation in different forms. Now, while proving we assume that the gas is ideal. For real gasses the expression changes a bit and some correction terms are added.