4.4 The Ideal Gas Law
We have just seen that the volume of a specified amount of a gas at constant pressure is proportional to the absolute temperature. In addition, we saw that the volume of a specified amount gas at a constant temperature is also inversely proportional to its pressure. We can correctly assume that pressure of a specified amount of gas at a constant volume is proportional to its absolute temperature. Let us also add the fact that the volume at constant pressure and temperature is also proportional to the amount of gas. Similarly, the pressure at constant volume and temperature is proportional to the amount of gas. Thus, these laws and relationships can be combined to give Equation 4.10.
Equation 4.10
Here, n is the number of moles of gas: Again, an absolute temperature must be used along with an absolute pressure.
Scientists and engineers have defined an ideal gas to be a gas with properties affected only by pressure and temperature. Thus, Equation 4.10 only needs a magical constant so that any one of its variables can be calculated if the other three are known. That constant is the ideal gas constant R and is used to form the Ideal Gas Law given by Equation 4.11.
Equation 4.11
Depending on the units of measure for the pressure, the volume, the number of moles, and the absolute temperature, some values for the ideal gas constant R are given in Table 4.1 and Appendix C for different units-of-measure systems.
Table 4.1 Values for R, the Ideal Gas Constant
System |
Value with Units of Measure |
SI |
8.314 (kPa)(m^{3})/(kg mol)(°K) |
cgs |
0.08206 (L)(atm)/(g mol)(°K) |
English |
0.7302 (ft^{3})(atm)/(lb mol)(°R) |
English |
10.73 (ft^{3})(psia)/lb-mol)(°R) |
The volume determined in the previous example is important and is usually memorized by engineers. It is the unit volume for 1 lb mol of gas at 32°F and 1 atm or 14.7 psia, which is usually expressed as 359 ft^{3}/lb mol. Besides the standard volume of 1 lb mol at 32°F and 1 atmosphere, another standard unit volume primarily used by the natural gas industry is defined at 59°F and 1 atmosphere with the value of 379 ft^{3}/lb mol.
There are several ways that Equation 4.11 can be used. The most obvious is to “plug” all of the numbers into the equation each time there is a change and “turn the crank” to calculate a new volume or pressure, or whatever is unknown. However, most of the time not all of the variables are known for this method or it involves a lot of unnecessary arithmetic. In addition, it also involves looking up and confirming one of the gas constants previously described even if they have been memorized.
We could also go back to Boyle’s Law and Charles’s Law if we were changing only one variable, but due to thermodynamic effects we will discuss in the next chapter, the temperature of a system will usually change whenever the pressure changes.
So, what is the easiest way? Simply use Equation 4.11 to make a ratio of the variables before and after; this gives us Equation 4.12.
Equation 4.12
The subscripts 1 and 2 indicate the before and after states, respectively. Notice that the ideal gas constant R has been canceled because its ratio is unity. We can rewrite Equation 4.12 to calculate a new pressure when a given quantity of gas is compressed and it becomes Equation 4.13.
Equation 4.13
Alternatively, we can rewrite it to calculate the effect on the volume when a gas is compressed to a new pressure and temperature, as shown in Equation 4.14.
Equation 4.14
We could arrange Equations 4.13 and 4.14 to calculate the ratio of the pressures or volumes after and before. That would give us Equation 4.15.
Equation 4.15
We could also rewrite Equations 4.13 and 4.15 to give us Equation 4.16.
Equation 4.16
We can arrange Equation 4.12 as needed to calculate a final pressure, volume, or temperature given the before-and-after states of all of the other variables. Assuming that we understand the Ideal Gas Law and the “PVT” relationship between pressure, volume, and temperature, it is a lot easier to remember just one equation and rearrange it as necessary.
In addition to calculating absolute values for pressure, volume, and temperature, as stated above we can calculate ratios that may be more useful. This is demonstrated in Example 4.4.
Not only are relative changes affected by temperature and pressure, but they are also affected by a change in the number of moles of gas when a chemical reaction occurs. Recently, the construction of a number of ethylene crackers has been announced in the Gulf Coast area due to the abundance of natural gas that contains ethane. Those crackers use steam with the ethane in a complex set of chemical reactions that reduce the amount of solid carbon formed, consequently reducing the cracker’s “coking” caused by the deposition of elemental carbon. In those crackers, 1 mole of ethane will produce 1 mole of ethylene and 1 mole of hydrogen according to the chemistry of:
C_{2}H_{6} → C_{2}H_{4} + H_{2}
An ethylene cracker is essentially a high-temperature furnace in which ethane flows through the piping, reaching temperatures on the order of 1,500°F in a matter of a few milliseconds. At that temperature, ethane starts to “pyrolyze” and form ethylene with a double bond as two hydrogen atoms are literally broken off. After exiting the furnace, it is quenched almost immediately lest it continue to break down, eventually becoming elemental carbon and molecular hydrogen.
The preceding example was actually a simple problem that has been made more complicated here. The simple solution can provide an estimate for Step 13 of the problem-solving technique in Chapter 1, “Introductory Concepts,” in which we judge our results. First, the cracking of ethane into ethylene with the hydrogen being given off doubles the number of moles of gas. Second, the increase of temperature, from 810°R (350°F) to 2,010°R (1,550°F), also more than doubles the volume. Thus, the volume of the gases would be about quadrupled, which approximates and confirms the 4.7 times increase in our previous calculation.
The exact increase in velocity is needed when designing the ethane cracker to insure that there will be sufficient residence time. Also, if the residence time is too great, the ethylene product will continue to crack to carbon and hydrogen gas. However, the actual calculations of the effect on cracking are much more difficult as the increase of the gas volume and thus the velocity is over the length of the pipe, while the temperature increase that more than doubles the volume occurs near the start of the pipe.