What type of substances allows the use of the Ideal Gas Law?

Question

I know that I can use the ideal gas law with pure gases or pure liquids. But can I also use the ideal gas law at saturated gases and saturated liquids as long as they aren't two phase substances?

Answer 1

The ideal gas law is derived from a model (the ideal gas), and like every other model it applies where it's underling assumptions are good approximations to reality.

So, important assumptions for the idea gas law:

• Point particles In the ideal gas, the particles occupy no volume. A real gas in which the atoms of molecules occupy a vanishing fraction of the volume is a good approximation. This suggests low density. Note that most liquids decidedly do not qualify.

• Non-interacting Philosophically the non-interaction assumption is a little bit difficult when you go to treat the ideal gas in thermodynamics and especially in statistical mechanics. I will suggest that having no internal degrees of freedom that are available for excitation at the energy of the gas and no significant interactions at ranges comparable to the average inter-particle distance qualifies. Because the accessible energy levels scales as the temperature of this gas, we should perhaps require moderate temperatures. Mono-atomic gasses will be be a good approximation to higher energies than more complex molecules (which generally have rotation and vibrational modes at lower energies than the atomic electron excitations).

• Random motion Situations where the other conditions apply and this one fails are rare, so I'm going to skip it.

So, what happens if this assumptions are violated? Well, the Van der Waals gas for the space occupied by the molecules and a bulk attractive force between molecules. This makes it applicable to higher density materials (but still ones whose internal degrees of freedom are not excited) and causes it to exhibit the gas-to-liquid first-order phase change (which is not present in the ideal gas).

Answer 2

dmckee gives some good qualitative considerations, but we can also develop rules for when the ideal gas law is and isn't appropriate. To start:

• The law applies perfectly in the case of a gas when $P\rightarrow 0$.
• The law does not apply to liquids.

Between these two states is a gray area. In that case you should look at the compressibility factor, $Z=P_\text{actual}/P_\text{ideal}$. $Z$ is a function of reduced pressure $P_r$ and reduced temperature $T_r$ (more on these later), and this correlation is given in standard charts which apply for most substances (I use one from Koretsky 2004, p. 198). If you accept errors up to 10%, you may apply the ideal gas law as long as $0.9<Z<1.1$. So:

• The law is a good approximation when $P_r<0.1$ (even for a saturated vapor).
• The law is a good approximation when $0.1<P_r<7$ if $T_r>1.819-\dfrac{0.3546}{P_{\!r}^{\,0.6}}\,$.
• The law is not a good approximation when $P_r>7$, no matter the temperature.

$P_r$ is defined as $P/P_c$ and $T_r$ is defined as $T/T_c$, where $P_c$ and $T_c$ are the substance's critical properties. For pure substances, these can be looked up in tables. For mixtures of vapors and gases which don't interact strongly, calculate each by multiplying the critical property of each pure component with its volume fraction and adding them together.

For example, pure water has $P_c=217~\text{atm}$ and $T_c=647~\text{K}$. Pure water vapor at 1 atm and 373 K has $P_r=1/217=0.0046$, so the ideal gas law applies to within 10% error. Pure water vapor at 25 atm and 498 K has $P_r=0.12$ and $T_r=0.77$, and $$0.77\not>1.819-\frac{0.3546}{0.12^{0.6}}$$ Thus the ideal gas law is no longer a good approximation. But if the vapor is mixed with 80% air $(P_c=37~\text{atm},\ T_c=133~\text{K})$ and kept at the same total pressure, we get $$P_c=80\%\cdot 37+20\%\cdot 217=73\Rightarrow P_r=0.34$$ $$T_c=80\%\cdot 133+20\%\cdot 647=236\Rightarrow T_r=2.1$$ $$2.1>1.819-\frac{0.3546}{0.34^{0.6}}$$ So the ideal gas law applies again.

But these rules only apply if you accept errors up to 10%. If accuracy is important, only use the ideal gas law for $P_r<0.025$ and don't use it for saturated vapors at all. When the ideal gas law doesn't apply, correct it using the compressibility factor $(P_\text{actual}=ZP_\text{ideal})$ or use a better equation of state like Soave-Redlich-Kwong or Peng-Robinson (not van der Waals; it's bad for general use).

Answer 3

Aside from the classical cases where the ideal gas law applies, it also applies to describe the exact entropy of a dilute solution, even if that solution is in a dense liquid. The reason is that the entropy of a dilute solution in a dense liquid is exactly the same as the entropy of a dilute gas, the number of possible positions for the solute particles is the same as the number of possibilities for the gas.