Real Gases
Thought Question: How does the volume of one mole of methane gas (CH4) at 300
Torr and 298 K compare to the volume of one mole of an ideal gas at 300 Torr and 298
K?
a) the volume of methane is greater than that of the ideal gas
b) the volume of methane is less than that of the ideal gas
c) the volumes are the same
Model 1: The Ideal Gas
Piston
Weight
P = gas pressure (atm)
n is number of moles of gas
R is the gas constant 0.08206 L•atm•K-1•mol-1
T is gas temperature in Kelvin
V is volume of the container
PV = nRT
1. What is the value of the quantity PV for one mole of an ideal gas at 25.0°C and one
atm?
2. What is the value of the quantity PV for one mole of an ideal gas at 25.0°C and
0.100 atm?
3. What is the value of the quantity PV for one mole of an ideal gas at 25.0°C and 100
atm?
4. On the graph below, sketch PV vs P for one mole of an ideal gas at 25.0°C
PV (L•atm)
P (atm)
5. If P=1.00 atm and T=25.0°C what is the volume of 1.00 mole of an ideal gas?
Information
The compressibility, Z, of a gas is defined as follows:
Z=
PV
where V is the molar volume (volume of 1.00 mole of gas)
RT
6. What are the units of Z?
7. What is the value of Z for an ideal gas?
8. On the graph below make a sketch of Z vs P for one mole of an ideal gas at 25°
Z
P (atm)
Information: The compressibility factor is a measure of how much a real gas
deviates from ideality
Note: in this figure, pressure
is shown in bars. The shape
of the curve is the same as it
would be if plotting in atm.
9. Add a sketch of Z vs. P for an ideal gas at 298 K to the graph in Figure 1. Explain
why you drew the graph as you did.
10. As pressure approaches zero, what is Z for a real gas?
11. Under what conditions of pressure do real gases behave most ideally? Explain why
this is true.
12. If Z > 1, is V
greater than, less than, or equal to Videal ? Explain your reasoning.
Model 2: Real Gases
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We have assumed that particles of an ideal gas are “point
masses” with zero volume. Real gas molecules have
volume. The “excluded” volume per mole of is given by b.
This is the volume occupied by the gas molecules
themselves. Here we are assuming that the molecules
are not interacting with one another.
Critical Thinking Questions
nRT
. For given values of n, R, V and T,
V
will the pressure of the real gas shown in Model 2 be smaller, the same, or larger
than the value of P calculated from the ideal gas equation? Explain.
12. For an ideal gas, we have seen that P =
nRT
. How could this equation be modified to correct for the
V
fact that real gas molecules have volume?
13. For an ideal gas, P =
14. For methane, b=0.0428 L/mol. Use the equation you derived in question 15 to
calculate the values of P, PV, and Z for one mole of methane at 25°C and the
volumes below (fill in the missing values):
15. In the table above, are the PV values higher, lower, or the same as those calculated
for an ideal gas? Explain.
Volume
(L)
Ideal
Pressure
(atm)
True
Pressure
(atm)
PV
(ideal)
PV
(real)
244.7
24.47
4.894
0.2447
0.1223
Model 3: Attractive Forces
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Molecules of real gases are attracted to one another. This attractive force
between molecules reduces the force when a molecule strikes the wall,
decreasing the pressure.
nRT
an 2
P=
V − nb
−
V2
b is the excluded volume per mole of gas
V-nb is the effective volume of the container
a is a measure of the strength of the attractive forces between molecules
Critical Thinking Questions
16. Why do you think that the van der Waal’s a value for H2O is greater than that of
CH4?
17. Assume that the volume of the container above is 24.789 L and there is one mole of
a real gas. At 25°C, will the real gas have a greater pressure or a lower pressure
than the gas in model 2? Explain.
18. The following graph shows a gas at several temperatures (notice that the x-axis
scale is quite large, so low pressures are near zero on the scale).
a. at what pressures does this gas tend to exhibit behavior closer to ideal gases
behavior?
b. At what temperatures does this gas tend to exhibit behavior closer to ideal gases
behavior?
c. Which of the following is correct?
i. Gases behave more ideally at high pressures and high temperatures
ii. Gases behave more ideally at low pressures and low temperatures
iii. Gases behave more ideally at low pressures and high temperatures
iv. Gases behave more ideally at high pressures and low temperatures
19. In general, Z tends to be larger than 1.00 at very high pressures (greater than about
350 atm for methane). Which van der Waals factor,
a or b seems to be most
responsible for this behavior? Explain.
20. In general, Z tends to be smaller than 1.00 at moderate pressures (around 150 atm
for methane). Which van der Waals factor,
this behavior? Explain.
a or b seems to be most responsible for