Adiabatic approximation in the derivation of the speed of sound

[Terry Bing]

The speed of sound in a gas at temperature T is given to be $v=\sqrt{\frac{\gamma RT}{M}}$, where $\gamma$ is the adiabatic exponent, R is the gas constant and M is the molar mass of the gas. In deriving this expression, we assumed that the compression and expansion processes were so fast that there isn't enough time for heat transfer to take place, and therefore that the processes are approximately adiabatic (Laplace's correction to Newton's formula). We therefore used $PV^\gamma = constant$. However the relation $PV^\gamma = constant$ is valid only for a quasi-static adiabatic process, and cannot be used for very fast processes. The quasi-static assumption is made in the derivation of adiabatic process relations. So how are we allowed to use it in the speed of sound derivation? The very reason we assumed the process was adiabatic is because it is very fast in the first place. So how is it consistent to assume it is quasi-static as well?

 

[Herman Trivilino]

The process has to be slow enough that it's quasi-static, but fast enough that negligible heat is transferred.

 

[Terry Bing]

The process has to be slow enough that it's quasi-static, but fast enough that negligible heat is transferred.

How do we know that this case satisfies that condition? Is there a practical limit below which a process can be reasonably assumed to be quasistatic? I have read that we can make the quasistatic approximation for expansion and compression processes, if the boundaries are moving much slower than the speed of sound in the gas. This makes perfect sense because then there would be sufficient time for the boundary disturbance to propagate through the rest of the gas and equalize for each small part of the process. However, I don't see how I can use that here, since the very derivation is for speed of sound in the gas.

 

[Herman Trivilino]

How do we know that this case satisfies that condition?

It's an assumption used in the derivation. If the result of the derivation matches what we observe then that gives us confidence that the assumption is justified.

 

[mfig]

FWIW, my text derives the basic relation ($c = \sqrt{\frac{\Delta p}{\Delta \rho}}$) using mass and momentum equations, then justifies the use of the isentropic relation ($\frac{p}{\rho ^k} = const$) by referring to experimental results which "indicate that the relationship between pressure and density across a sound wave is nearly isentropic."

So, rather than being an assumption in the derivation, it is an experimental finding that is applied to the general result.

 

[Terry Bing]

It's an assumption used in the derivation. If the result of the derivation matches what we observe then that gives us confidence that the assumption is justified.

Thanks. I get that it matches the observed speed of sound. I am just curious as to why? Was there a way we could have guessed beforehand that it would work?

 

[vanhees71]

You can derive hydrodynamics from the Boltzmann-transport equation. The ideal hydrodynamics equation, the Euler equation of an ideal fluid, comes out when you consider the fluid being always in local thermal equilibrium, i.e., at any time the phase-space distribution function is that of an ideal gas with temperature and chemical potential being functions of time and position. This approximation is valid if the typical changes of these macroscopic quantities are slow in time and with not too large gradients in space. This implies that no heat is transferred and no dissipation occurs, i.e., the thermodynamic process is adiabatic, i.e., entropy doesn't change. That's because the collision term of the Boltzmann equation then vanishes.

On top of this ideal-hydro approximation you can consider deviations from local thermal equilibrium using a gradient expansion or the method of moments. This leads to the calucation of transport coefficients like heat conductivity and shear and bulk viscosity. In the lowest order of the gradient expansion you obtain the Navier-Stokes equation.