Pipes, resonating frequencies and yeah some gases

[mooncrater]

Homework Statement

A closed organ pipe resonates in its fundamental mode at a frequency of $200Hz$ in $O_2$ at a certain temperature. If the pipe contains 2 moles of $O_2$ and 3 moles of $O_3$ are now added to it, then what will be the fundamental frequency of same pipe at same temperature?
[Given answer is $172. 7Hz$]

Homework Equations

The relevant equation according to me is:
$v =√(\gamma P/\rho)$ (in a gas speed of a wave)

The Attempt at a Solution

What I did is:
Velocity of a wave in a gas=
$$v=√(\gamma P/\rho)$$
So using $v=\nu\lambda$
We can say that
$\nu_1/\nu_2=√(\rho_2/\rho_1)$
And we know that
$\rho_1=4×16/V$ where V is the volume of pipe
And $\rho_2=3×3×16+2×2×16/V=13×16/V$
Therefore my frequency is coming out to be
$110 Hz$
Which is wrong.. I know the whole of this seems to be wrong from the start... so how to do this?

 

[BvU]

Are you assuming P and $\gamma$ remain the same ?

 

[Raghav Gupta]

Homework Statement

A closed organ pipe resonates in its fundamental mode at a frequency of $200Hz$ in $O_2$ at a certain temperature. If the pipe contains 2 moles of $O_2$ and 3 moles of $O_3$ are now added to it, then what will be the fundamental frequency of same pipe at same temperature?
[Given answer is $172. 7Hz$]

Homework Equations

The relevant equation according to me is:
$v =√(\gamma P/\rho)$ (in a gas speed of a wave)

The Attempt at a Solution

What I did is:
Velocity of a wave in a gas=
$$v=√(\gamma P/\rho)$$
So using $v=\nu\lambda$
We can say that
$\nu_1/\nu_2=√(\rho_2/\rho_1)$
And we know that
$\rho_1=4×16/V$ where V is the volume of pipe
And $\rho_2=3×3×16+2×2×16/V=13×16/V$
Therefore my frequency is coming out to be
$110 Hz$
Which is wrong.. I know the whole of this seems to be wrong from the start... so how to do this?

First what is the gamma for diatomic gas?

 

[mooncrater]

Okay, $\gamma$ Will also change for $O_2$ and $O_3$.
For a diatomic gas it is 7/5 and for a triatomic gas it's 4/3.

 

[mooncrater]

Thanks I got it now​

 

[BvU]

Care to enlighten us with your workings ?

 

[mooncrater]

Care to enlighten us with your workings ?

Yup... why not...
I just put the values of $\gamma$
too, instead of cancelling them while comparing the resonant frequencies in the two cases.

 

[Raghav Gupta]

Yup... why not...
I just put the values of $\gamma$
too, instead of cancelling them while comparing the resonant frequencies in the two cases.

But γ would not be 4/3 when you mix both of them.
What would be equivalent γ then?

 

[mooncrater]

Hmmm.. $\gamma$ can be calculated by comparing the internal energies of the gases in the two cases. :
$n_1C_v1ΔT+n_2C_v2ΔT=(n_1+n_2)C_{net}Δ T$
Cancelling $ΔT$
$2×(7/5)R+3×(4/3)R=5×R/(\gamma-1)$
Cancelling $R$
$\gamma=59/34$
But it's giving the answer 123. 51 Hz

 

[Raghav Gupta]

Hmmm.. $\gamma$ can be calculated by comparing the internal energies of the gases in the two cases. :
$n_1C_v1ΔT+n_2C_v2ΔT=(n_1+n_2)C_{net}Δ T$
Cancelling $ΔT$
$2×(7/5)R+3×(4/3)R=5×R/(\gamma-1)$
Cancelling $R$
$\gamma=59/34$
But it's giving the answer 123. 51 Hz

What about P?

 

[BvU]

I would expect $\gamma$ to come out between 1.4 and 1.3, not at 1.74