# How is ideal gas law applicable to real gas?

I read one of the questions here on physics.stack exchange proving how the speed of sound increases with temperature using ideal gas law equation and adiabatic index. Here's the link:

But how is the ideal gas law applicable to real gas? I'm sorry if this is an incredibly dumb question. I'm still in high school.

## 3 Answers

Ideal gas law works best for monatomic gasses at low pressures and high temperatures.

It doesn't take into account molecular size and intermolucular interactions, so when the effects of those are significant, then the equation will be less accurate.

It generally works well for gasses like air in the temperatures and pressures we are used to (air has a lot of diatomic gasses which don't interact highly with each other and behave fairly ideally at room temperature).

Wikipedia has a fairly good entry on it here.

The speed of sound is defined as $c^2 = \frac{\partial p}{\partial \rho}$, which for an ideal gas becomes $c^2 = \gamma \frac{p}{\rho}$.

For a real gas, the relationship to an ideal gas can be found through the compressibility factor $z$. This is a measure of how much a real gas deviates from an ideal gas. It shows up in the equations as:

$$P = z \rho R T$$

You can work through the math on it (or follow along on this page), but essentially for a real gas, the speed of sound uses the compressibility factor $z$ and a factor $n$:

$$n = \gamma \frac{z + T \partial z/\partial T \rvert_p}{z + T \partial z/\partial T\rvert_\rho}$$

such that:

$$c^2 = zn \frac{p}{\rho}$$

which can be related to the ideal gas speed of sound as:

$$c^2_{real} = z c^2_{ideal} \frac{z + T \partial z/\partial T \rvert_p}{z + T \partial z/\partial T\rvert_\rho}$$

For gases and conditions where intermolecular forces are unimportant, $z = 1$ and $\partial z/\partial T \approx 0$ and the ideal speed of sound is correct. This is true for a majority of gasses at ambient conditions. At very high pressures and/or low temperatures, real gas effects become important.

• I'll add that the $P = z \rho RT$ equation is still a fairly simplified version of real gas laws, which generally have much messier equations and are all different depending on which model you use and how it's applied. – JMac Mar 13 '17 at 12:08
• @JMac Indeed. I'm familiar with the Peng-Robinson and Redlich-Kwong, but I know there are many more. But given the background in the question, I don't know if it was worth really digging into cubic EOS's and more complex models at this stage. – tpg2114 Mar 13 '17 at 12:10
• Well he was wondering how an ideal gas law relates to a real gas. This equation helps to show an ignored factor, but the real gas equations help to illustrate how much is really not considered when we use the ideal gas law. The compressibility factor glosses over all the complexity of what we ignore by lumping it in a single term. I just think it's probably worth mentioning if you're showing the compressibility equation anyways. – JMac Mar 13 '17 at 12:13

Speed of sound = sqaure root of pressure upon density now here let the gas be ideal or real the pressure is going to increase with rise in temperature so this law can be used here

• This is very unclear about answering the question. – JMac Mar 13 '17 at 12:38

## protected by Qmechanic♦Mar 13 '17 at 14:36

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