Answer
64.8k+ views
Hint: Ideal gas is a hypothetical gas whose molecules occupy negligible space and have no interactions, and which consequently obeys the gas laws exactly. The speed of sound in a gas can be calculated as the square root of (the coefficient ratio of specific heats × the pressure of the gas / the density of the medium).
Formula used:
$PV = nRT$ , $v = \sqrt {\dfrac{{\gamma RT}}{M}} $ , $v = \sqrt {\dfrac{{\gamma P}}{\rho }} $
Complete step by step solution:
From kinetic theory of gas, we can write
$v = \sqrt {\dfrac{{\gamma RT}}{M}} $
Thus velocity of the sound in gas is directly proportional to square root of temperature
That is, proportional to $\sqrt T $
Also, from the ideal gas law, also called the general gas equation
$PV = nRT$ (Here taking $n = 1$ , as considering 1 mole of gas)
Therefore $PV = RT$
Substituting the value of $RT$ in $v = \sqrt {\dfrac{{\gamma RT}}{M}} $$\left( M \right)$
We get $v = \sqrt {\dfrac{{\gamma PV}}{M}} $
$v = \sqrt {\dfrac{{\gamma P}}{\rho }} $ (As $\rho $ is the density of the gas as $\rho = \dfrac{M}{V}$ )
Thus velocity of the sound in gas is directly proportional to square root of pressure when density of gas is constant.
From the given formula it is clear that the velocity of the sound in the gas is proportional to $\sqrt T $ and $\sqrt P $ when $\rho $ is constant.
Note: For a given gas, $\left( \gamma \right)$ , gas constant $\left( R \right)$ and molecular mass $\left( M \right)$are constants. Then speed of sound depends only on temperature. It is independent of the pressure. But for a given gas, if adiabatic index $\left( \gamma \right)$ and density $\left( \rho \right)$are constants, the speed of sound depends only on the square root of pressure. It is independent of temperature.
Formula used:
$PV = nRT$ , $v = \sqrt {\dfrac{{\gamma RT}}{M}} $ , $v = \sqrt {\dfrac{{\gamma P}}{\rho }} $
Complete step by step solution:
From kinetic theory of gas, we can write
$v = \sqrt {\dfrac{{\gamma RT}}{M}} $
Thus velocity of the sound in gas is directly proportional to square root of temperature
That is, proportional to $\sqrt T $
Also, from the ideal gas law, also called the general gas equation
$PV = nRT$ (Here taking $n = 1$ , as considering 1 mole of gas)
Therefore $PV = RT$
Substituting the value of $RT$ in $v = \sqrt {\dfrac{{\gamma RT}}{M}} $$\left( M \right)$
We get $v = \sqrt {\dfrac{{\gamma PV}}{M}} $
$v = \sqrt {\dfrac{{\gamma P}}{\rho }} $ (As $\rho $ is the density of the gas as $\rho = \dfrac{M}{V}$ )
Thus velocity of the sound in gas is directly proportional to square root of pressure when density of gas is constant.
From the given formula it is clear that the velocity of the sound in the gas is proportional to $\sqrt T $ and $\sqrt P $ when $\rho $ is constant.
Note: For a given gas, $\left( \gamma \right)$ , gas constant $\left( R \right)$ and molecular mass $\left( M \right)$are constants. Then speed of sound depends only on temperature. It is independent of the pressure. But for a given gas, if adiabatic index $\left( \gamma \right)$ and density $\left( \rho \right)$are constants, the speed of sound depends only on the square root of pressure. It is independent of temperature.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)