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?
[Physics] What type of substances allows the use of the Ideal Gas Law
ideal-gasthermodynamics
Related Solutions
According to the second law, thermal equilibrium between two systems means that they both have the same temperature $T$. The fact 2 that $PV$ coincide whenever two gases are at thermal equilibrium (and, I assume, for the same $n$) means that $PV$ is only a function of $T$. In othor words, there is a function $g()$ such that $$\begin{align}PV&=g(T)& P=\frac{g(T)}{V}\end{align}$$
The goal is now to show that $g()$ is linear, i.e. that $g(T)=Tg'(T)$. In order to show that, one can use a Maxwell relation, more specifically, the one which is linked to Helmholtz free energy $A=U-TS$ which can be defined because of condition 3. We have then (including condition 4 for the computation of the derivatives) $$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V = -\frac{\partial^2 A }{\partial T \partial V}$$
From the first equation, we have $$\left(\frac{\partial P}{\partial T}\right)_V = \frac{g'(T)}{V}.$$
Except if $PV=g(T)$ is constant over a range of temperatures (which can be checked experimentally), the condition 5 implies that $U(T,V)=U(T)$. If one make as mall isotherm transformation of our gas, one has $$\begin{align} 0=dU&=\delta Q - PdV& \delta Q=PdV\end{align}.$$ The second principle tells us then that $$dS=\frac{\delta Q}{T}=\frac{P dV}T=\frac{g(T)}{TV}dV.$$ By definition, in this case $dS=\left(\frac{\partial S}{\partial V}\right)_TdV$, so we get this partial derivative from the last equation.. Equating the two partial derivatives according to the Maxwell equation then gives us $g(T)=T g'(T)$, which implies $g'(T)=nR$, where $nR$ is an "arbitrary" constant. Hence, $$PV=nRT.$$
In gases, under normal conditions, the average distance between molecules is large compared to size of the molecules so the molecules spend most of their time far apart. Interactions between molecules, or at least strong interactions between molecules, tend to be short range. This means that interactions between molecules don't have much effect on the overall properties of the gas because the molecules spend most of the time too far apart to interact strongly with each other. This means that to a good approximation we can ignore interactions between molecules completely, and this is what we mean by an ideal gas. An ideal gas is one where the gas particle do not interact.
So we don't have to worry whether the gas is made up from helium, or oxygen, or chlorine or whatever. As long as the molecules spend most of their time far apart the gas will behave as an ideal gas so all gases behave pretty much the same way. All we need to do is correct for the mass of the gas molecules, which is a straightforward numerical factor.
The trouble is that in liquids and solids the molecules are very close together. In fact in solids they are touching and in liquids they aren't far off that. Because the molecules are so close the interactions between them are strong, and therefore can't be ignored. That's why, for example, solid iron is very different to solid paraffin wax. Since we can't ignore interactions in liquids and solids there can be no universal ideal liquid or ideal solid equations that all liquids and solids obey.
Best 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:
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:
$P_r$ 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).