Speed of sound in a real gas

Question

The speed of sound in a real gas is $C = \sqrt{ZkRT}$, where $C$ is the speed of sound in the gas, $k$ is adiabatic gas constant $={C_p}/{C_v}$, $R$ is the gas constant and $T$ is gas temperature in Kelvin and $Z$ is compressibility factor.

Suppose we compress a real gas in an isothermal process ($T=$ constant). By increasing the pressure, we have a denser gas, so I would expect the speed of sound to increase.

On the other hand, for $T=$ constant, according to $C = \sqrt{ZkRT}$, speed of sound is proportional to $\sqrt{Z}$. According to marked up region in diagram below ($Z$ vs ${P_r}$ for ${T_r}$: constant) as pressure increases, $Z$ decreases, which means speed of sound decreases ($C\alpha\sqrt{Z}$).

In other words, according to the formula above and the graph below, by increasing pressure (in the marked up region), speed of sound decreases, which is against common sense (denser gas should have higher speed of sound). How to solve this paradox?

Answer 1

The speed of sound in an ideal gas is given by:

$$ v = \sqrt{\gamma\frac{P}{\rho}} \tag{1} $$

where $P$ is the pressure, $\rho$ is the density and $\gamma$ is the heat capacity ratio. The pressure is telling us about the stiffness of the gas i.e. the higher the pressure the more the gas resists being compressed or expanded so the higher the velocity. The density is telling us about the mass of the gas i.e. the higher the density the more force it takes to accelerate the gas so the slower the velocity.

In this form it isn't obvious what happens when we change the pressure because changing the pressure also changes the volume. To make the behaviour clearer we can substitute for $P$ using the ideal gas equation of state:

$$ PV = RT \tag{2} $$

and this gives us:

$$ v=\sqrt{\gamma\frac{RT}{M}} \tag{3} $$

where $M$ is the molar mass. Since $\gamma$, $R$ and $M$ are constants equation (3) tells us that if we keep the temperature constant the velocity of sound is constant. That is:

For an ideal gas at constant temperature the velocity of sound does not depend on the pressure

What your diagram is telling you is that the gas is not ideal. At low pressures the attractions between the molecules dominate and the the gas is "floppier" than an ideal gas, which makes the velocity of sound lower than an ideal gas.

This difference increases as we increase the pressure, hence the negative gradient in your graph. However at some point the molecules get close enough that the hard core repulsive interactions between molecules becomes important and from this pressure the gas starts getting "stiffer" again, which increases $v$ again.

Answer 2

Actually, the speed of sound increases with pressure. I guess you did not take into account the change in $k$ with pressure. I found a paper titled "Evaluation of Speed of Sound and Specific Heat Capacities of Real Gases", downloadable from researchgate . net (published in 2018). I will quote the values he got from E W Lemmon et al in 2010 "Thermodynamic Properties of Air and Mixtures of Nitrogen, Argon and Oxygen ..."

Speed of Sound

Pressure (atm) T=300K T=600K

1.000000----- 347.4------ 487.1

4.93462------- 347.8 ----- 488.0

9.86923------- 348.4----- 489.2

19.7385-------- 349.9----- 491.5

(dashes added to line up the table) Other temperatures are in the original.

Another reference I found gave an equation to relate $c_p-c_v$ in terms of $Z$:

From page 42 of masters thesis "Real Gas Thermodynamics" at Delft University of Technology. Using this with definition of $k$ the value of $k$ as function of $Z$ and $T$ could be found.