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:
How can the speed of sound increase with an increase in temperature?
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.
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.
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
