The Ideal Gas Law and the Steam Dome

The ideal gas law describes the approximate behavior of gases at temperatures that are high relative to the critical temperature and at pressures that are low relative to the critical pressure.  This figure shows the approximate region where the ideal gas law applies best. 

Normally, one would use steam tables to determine properties close to the saturation region, and only use the ideal gas approximation far away, as shown in the figure. Today we’ll look at the error introduced by the ideal gas approximation as we get close to the saturation region.

The ideal gas law can be written as:
P*v=R*T
where P is the pressure, v is the specific volume, R is the gas constant for a particular gas, and T is the temperature.
Using the ideal gas law, if we know P and T, we can calculate the specific volume, or if we know P and v, we can calculate the temperature.  If we are close to the saturation region, we will also have P-v-T data from the steam tables, and we can compare the calculated values of v, or T, to the tabulated values and see the error in the ideal gas approximation.

This figure shows a comparison of the ideal gas law approximation with the actual data (from the steam tables) for the saturated vapor line for water. 

The red line represents the saturated vapor line (the right side of the steam dome).  The blue line represents the calculation of v from values for temperature and pressure from the steam tables.  The green line represents the calculation of T from values for specific volume and pressure from the steam tables.
Clearly, the ideal gas approximation works better at low pressures (large specific volume) at the right of the figure.  Note that this entire comparison is along the saturated vapor line—where the vapor is just on the verge of starting to condense into liquid.  This is not where one would normally think of using the ideal gas approximation at all.

The next two figures provide a more quantitative look at the comparison. 

The percent error in both the v(T) and T(v) ideal gas calculations are plotted as functions of the pressure.  The second figure is just a zoomed-in view of the left side of the first figure.  Clearly, the ideal gas approximation does not work well for the majority of the saturated vapor line, as we would expect.  However, the second figure shows that at low enough pressures, the ideal gas approximation is reasonably accurate, even on the saturated vapor line.

Next time, we’ll look at the accuracy of using the ideal gas approximation in the superheated region.