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

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

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

• The law is a good approximation when $P_r<0.1$ (even for a saturated vapor).


• The law is a good approximation when $0.11.819-\dfrac{0.3546}{P_{\!r}^{\,0.6}}\,$.


• 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).