Abstract, Background, Introduction
The Ideal Gas Law is very useful in the characterization of gases:
The Ideal Gas Law has two assumptions, that the molecules are “point-like” making the volume of the molecules insignificant, and there are no interactions between the molecules. There are limitations to the Ideal Gas Lae and it experiences error at increases in pressures or temperature. This deviation is due to the fact that the more interactions between molecules occur at higher temperatures and pressures.
The compressibility coefficient (Z) is used to assess the validity of the Ideal Gas Law.
For an Ideal gas the compressibility factor coefficient equals 1. With the exception of noble gases and hydrogen gas, the compressibility coefficient is usually less than 1 at low pressures and higher than 1 at high pressures.
There has been some attempt to account for the deviations from the ideal gas law. One example is the van der Waals equation, but there are still limitations. Another approach to characterize “real gases” is to determine the deviation of the compressibility coefficient from ideal behavior by observing it as a function of pressure in the form of a virial equation.
B2,B3,B4, are the second, third, and forth virial coefficients. The more deviation from ideal behavior, the more accuracy that is needed by considering higher terms for the virial equation.
For this particular experiment the deviation from ideal behavior will not be as significant because the pressure of the gas will be between 1-35 bar and therefore only the second virial coefficient is needed. The second form of the equation is used for this experiment.
A2’s relationship to B2 can be expressed using the following equation:
There is yet another approach to measuring the deviation from the ideal behavior and it is the determination of the fugacity coefficient.
f is the fugacity, which is the experienced pressure of the gas and γ is the fugacity coefficient.
The goal of this experiment is to gain information to characterize the gas (CO2) being tested. The sample tank must be loaded with gas at 35 bar. Then release a small amount of gas into the expansion tank. By monitoring the pressure of the two tanks the amount of sample gas that was released can be calculated using the ideal gas law. Evacuate the explosion tank, release another small sample of gas from the sample tank into the evacuation tank. This process must be continued until the pressure in the two tanks is the same. The sum of the released samples is the initial number of moles of gas in the sample tank. The first sample gas release from the sample tank can be expressed using the following equation.
P1eis the pressure in the expansion tank after releasing the first sample. Ve is the volume of the expansion tank. In future calculations the pressure and volume of the sample tank is represented in the same fashion only the e is replaced with an s. The second sample of gas released can be represented by:
The same process can be used for each step and the following equation can be obtained to express the initial number of moles of gas.
The number of moles of gas after the rth release in the sample tank can be represented by:
Combining the last two equations yields:
Using substitution the following equations can be obtained for the compressibility factor:
For this experiment the pressure must be measured at every release of gas in both the explosion tank and the sample tank. Only the ratio of the two volumes of tanks is relevant. After the experiment is complete the sample tank is closed, the expansion tank is evacuated and all the gas will expand into the expansion tank and the pressure will be recorded as the final pressure (pf).
The ideal gas law can then be used to calculate the ratio of the tank volumes.