# ${\rm CO}_2$ Pressure Vs Temperature in Supercritical Phase

Say we have a compressed gas cylinder half full of liquid CO2 under its own vapor pressure. Below 31C the pressure and temperature in the cylinder follows the saturation line. If you know the temperature, you know the pressure. Easy.

However, above 31.2C, CO2 is supercritical and, as far as I can tell, you need more than just the temperature to know the pressure.

So, if we take the same cylinder and continue to heat it above 31C, what is the relationship between pressure and temperature? What would be the pressure in the cylinder at 40C? 50C?

• Below 31C an "exact solution" would also need to account for amount of gas in liquid vs gas phase, and the density change of the liquid phase. – MaxW Feb 28 at 20:26

You need to use the law of corresponding states for this problem, where the ideal gas law has been modified to read

$$PV=znRT$$

The term "z" is known as the compressibility factor. This factor is calculated based on the reduced pressure and reduced temperature of the substance in question, where

$$P_r=P/P_{critical}$$

$$T_r=T/T_{critical}$$

Much more information, and a generalized chart that can be used to determine the compressibility factor, can be found here: https://en.wikipedia.org/wiki/Compressibility_factor

The first thing to recognize is that there's a difference in "knowing the pressure" and "modeling the pressure." If the model is really good, then "modeling" does mean "knowing," but that can be hard or impossible to do sometimes. Below the transition point, the pressure is related to the temperature through the ideal gas equation of state -- the classic $$pV = nRT$$ or $$p = \rho R T$$. When you have a supercritical fluid, however, this equation of state doesn't work so well and it is no longer a good model.

Unfortunately, supercritical fluids are hard to model. Very hard to model. The most successful equations of state for them are called cubic equations of state. You can see that they add complexity to be sure, and they begin to introduce additional constants and non-linear functions. These constants are usually calibrated to experimental measurements and critical properties, but these have to be measured away from the critical points. Many thermodynamic terms are undefined at the critical points, making modeling very complicated there. Mixtures of gases are also very complicated to model well, because the equations of state for the individual components have different constants and mixing rules to combine them aren't universal.

More to the point, though, you said that you need to know more than just the temperature to find the pressure. This isn't the most precise way to say it -- yes, you do need to know additional constants and additional functional forms to relate pressure and temperature. But, no matter how complicated the equation of state becomes, any two state relations can be connected by a third! In all cases, if you have $$V,T$$ or $$\rho, T$$, then you can find $$p$$. It might be linear, like $$p = \rho R T$$, or it could be cubic like the more complicated equations I linked to. Or it could be some much more complicated functional model that hasn't been developed yet. But any two state variables can find the third.

• The "unadjusted" ideal gas law only works well at low pressures and high temperatures, which is not the case for the OP's problem, or for problems where the reduced temperature is relatively low or the reduced pressure is relatively high. For a look at "relatively", look at the deviation of "z" from unity on the chart in the Wikipedia article that I referenced. – David White Feb 28 at 20:35