
At a given temperature if is the root mean square velocity of the molecules of a gas and is the velocity of sound in it. Then these are related as
A.
B.
C.
D.
Answer
500.4k+ views
Hint:Here , the velocity of sound can be found out using the equation,
Where is the ratio of specific heat at a constant pressure to specific heat at constant volume.
P is the pressure and is the density
With the help of an ideal gas equation we can write this equation in terms of temperature and mass and universal gas constant.
Then on comparing this equation with the equation of root mean square velocity we can reach the final answer.
Complete step-by-step solution:
We need to find the relationship between root mean square velocity and the speed of sound at a given temperature.
We know that the root mean square velocity is given by the equation
(1)
Where, R is the universal gas constant, T is the temperature in Kelvin and M is the molecular weight.
We know that the velocity of sound can be found out using the equation,
(2)
Where, is the ratio of specific heat at a constant pressure to specific heat at constant volume.
P is the pressure and is the density.
We know that density is equal to mass divided by volume. That is,
Let us substitute this in equation 2 .
(3)
From the ideal gas equation, we have
Let us take the number of moles n to be one. Then we can write PV is equal to RT.
Substituting this in the equation 3 we get
(4)
Now let us divide equation 4 with equation 1.
Then we get,
Therefore,
This is the relation between RMS velocity and velocity of sound at a given temperature.
So the correct answer is option B.
Note:-
Alternative method –
Velocity of sound is given as .
Where is the density and P is the pressure.
The relation between pressure P and root mean square velocity is given as
On substituting this in the first equation we get
.
Where
P is the pressure and
With the help of an ideal gas equation we can write this equation in terms of temperature and mass and universal gas constant.
Then on comparing this equation with the equation of root mean square velocity we can reach the final answer.
Complete step-by-step solution:
We need to find the relationship between root mean square velocity and the speed of sound at a given temperature.
We know that the root mean square velocity is given by the equation
Where, R is the universal gas constant, T is the temperature in Kelvin and M is the molecular weight.
We know that the velocity of sound can be found out using the equation,
Where,
P is the pressure and
We know that density is equal to mass divided by volume. That is,
Let us substitute this in equation 2 .
From the ideal gas equation, we have
Let us take the number of moles n to be one. Then we can write PV is equal to RT.
Substituting this in the equation 3 we get
Now let us divide equation 4 with equation 1.
Then we get,
Therefore,
This is the relation between RMS velocity and velocity of sound at a given temperature.
So the correct answer is option B.
Note:-
Alternative method –
Velocity of sound is given as
Where
The relation between pressure P and root mean square velocity
On substituting this in the first equation we get
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