The Ideal Gas Law assumes that molecules occupy no volume and experience no intermolecular forces. Real gases do not satisfy either assumption. Molecules have finite size, and attractive and repulsive interactions occur between neighboring molecules. As a result, real gases deviate from ideal behavior, particularly at high pressures and low temperatures.
To account for these deviations, a number of more sophisticated equations of state have been developed. These equations attempt to describe how molecular size and intermolecular interactions affect the relationship between pressure, volume, and temperature.
| Model | Equation of State |
|---|---|
| Ideal Gas | \(PV=nRT\) |
| van der Waals | \[ \left( P+\frac{an^2}{V^2} \right) (V-nb) = nRT \] |
| Dieterici | \[ P(V-b) = RT \exp \left( -\frac{a}{RTV} \right) \] |
| Virial Equation | \[ \frac{PV_m}{RT} = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} +\cdots \] |
Each of these models attempts to account for deviations from ideal behavior, but they do so in different ways and with varying degrees of complexity.
One of the most important and historically significant real-gas models is the van der Waals equation:
\[ \left( P+\frac{an^2}{V^2} \right) (V-nb) = nRT \]
This equation modifies both the pressure term and the volume term of the Ideal Gas Law in order to account for the two major ways that real gases differ from ideal gases.
The correction
\[ \frac{an^2}{V^2} \]
accounts for attractive intermolecular forces. Molecules near the wall of a container are attracted by neighboring molecules in the gas, reducing the force with which they strike the wall. This causes the observed pressure to be lower than predicted by the Ideal Gas Law.
The parameter \(a\) therefore measures the strength of intermolecular attractions. Large values of \(a\) correspond to substances with strong attractive forces.
Examples:
The large value for CO2 reflects its relatively strong intermolecular attractions.
The correction
\[ V-nb \]
accounts for the fact that molecules occupy finite volume. The volume available for molecular motion is therefore smaller than the total volume of the container.
The parameter \(b\) is related to molecular size. Larger molecules occupy more space and therefore have larger values of \(b\).
Examples:
At high pressures, the finite size of molecules becomes increasingly important, causing significant deviations from ideal behavior.
Another useful approach is the Virial Equation of State:
\[ \frac{PV_m}{RT} = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} +\cdots \]
The virial equation expresses deviations from ideal behavior as a power-series correction to the Ideal Gas Law. The coefficient \(B(T)\), known as the second virial coefficient, contains most of the information about intermolecular interactions and is particularly important in the low-pressure limit.
When all virial coefficients are zero, the equation reduces to the Ideal Gas Law. As a result, the virial equation provides a convenient framework for understanding and quantifying deviations from ideality.
Big picture: Real gas equations of state extend the Ideal Gas Law by accounting for molecular size and intermolecular forces. The van der Waals parameters \(a\) and \(b\) provide a direct molecular interpretation of these effects, while the virial equation provides a systematic way to quantify deviations from ideal behavior.
Calculate the pressure exerted by \(1.00\ \mathrm{mol}\) of N2 at \(298\ \mathrm{K}\) in a \(24.4\ \mathrm{L}\) container using both the Ideal Gas Law and the van der Waals equation.
\[ PV=nRT \]
\[ P=\frac{nRT}{V} \]
\[ P= \frac{ (1.00\ \mathrm{mol}) (0.08206\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}) (298\ \mathrm{K}) } {24.4\ \mathrm{L}} \]
\[ P=1.00\ \mathrm{atm} \]
For N2, \(a=1.352\ \mathrm{atm\,L^2\,mol^{-2}}\) and \(b=0.0387\ \mathrm{L\,mol^{-1}}\).
\[ \left( P+\frac{an^2}{V^2} \right)(V-nb)=nRT \]
Solving for pressure:
\[ P= \frac{nRT}{V-nb} - \frac{an^2}{V^2} \]
\[ P= \frac{ (1.00\ mol)(0.08206\ \frac{atm\ L}{mol\ K})(298\ K) } { ((24.4\ L) - (1.00\ mol)(0.0387\ \frac{L}{mol})) } - \frac{ (1.352\ \frac{atm\ L^2}{mol^2})(1.00\ mol)^2 } { (24.4\ L)^2 } \]
\[ P=1.001\ \mathrm{atm} \]
Under these conditions, N2 behaves very nearly ideally. The ideal gas prediction is \(1.00\ \mathrm{atm}\), while the van der Waals prediction is only slightly larger.
The Ideal Gas Law provides an excellent description of gases under many conditions, particularly at low pressures and high temperatures. Real gases, however, are composed of molecules that occupy finite volume and experience attractive and repulsive intermolecular forces. As a result, real gases often deviate from the behavior predicted by the Ideal Gas Law.
A convenient way to quantify these deviations is through the compression factor, \(Z\), which compares the observed behavior of a gas to the behavior expected for an ideal gas.
\[ Z = \frac{PV_m} {RT} \]
where \(V_m\) is the molar volume of the gas.
For an ideal gas,
\[ PV_m = RT \]
and therefore
\[ Z = 1 \]
under all conditions. Real gases, however, often exhibit values of \(Z\) different from unity.
The value of \(Z\) provides information about the dominant intermolecular effects present in the gas.
| Compression Factor | Interpretation |
|---|---|
| \(Z=1\) | The gas behaves ideally. |
| \(Z<1\) | Attractive intermolecular forces dominate. |
| \(Z>1\) | Repulsive interactions and finite molecular size dominate. |
When attractive forces dominate, molecules pull one another inward, reducing the pressure exerted on the container walls. This causes the gas to occupy a smaller volume than predicted by the Ideal Gas Law, producing values of \(Z\) less than one.
At very high pressures, molecules are forced close together and their finite size becomes important. Under these conditions, repulsive interactions dominate and the gas occupies a larger volume than predicted by the Ideal Gas Law, producing values of \(Z\) greater than one.
Different gases approach ideal behavior to different degrees depending on the temperature. For a given gas, there is one special temperature known as the Boyle Temperature, \(T_B\), at which the gas behaves nearly ideally over a broad range of low pressures.
The Boyle Temperature can be defined mathematically as the temperature at which the compression factor becomes insensitive to pressure in the low-pressure limit:
\[ \left( \frac{\partial Z} {\partial P} \right)_T = 0 \]
Using the Virial Equation of State, this condition leads to
\[ B(T_B)=0 \]
where \(B(T)\) is the second virial coefficient.
At temperatures below the Boyle Temperature, attractive forces are more important and the gas tends to exhibit \(Z<1\). At temperatures above the Boyle Temperature, repulsive effects become more important and the gas tends to exhibit \(Z>1\).
The Boyle Temperature provides a useful measure of how closely a real gas approaches ideal behavior. Gases operated near their Boyle Temperature tend to follow the Ideal Gas Law over a wider range of pressures than they do at other temperatures.
The concept also provides insight into the balance between attractive and repulsive intermolecular forces. At the Boyle Temperature, these competing effects approximately cancel, making the gas appear ideal even though real intermolecular interactions are still present.
Big picture: The compression factor measures how much a real gas deviates from ideal behavior. The Boyle Temperature identifies the conditions under which a real gas most closely resembles an ideal gas, reflecting the balance between attractive and repulsive intermolecular interactions.
The Ideal Gas Law predicts that a gas can be compressed indefinitely by increasing the pressure. Real gases, however, behave quite differently. At sufficiently high pressures and low temperatures, gases can undergo a phase transition and condense to form liquids.
The behavior of a real gas is often visualized using an isotherm, a plot of pressure as a function of volume at a constant temperature. At high temperatures, the isotherms resemble the behavior predicted by Boyle's Law. At lower temperatures, however, the isotherms begin to show dramatic deviations from ideal behavior as condensation becomes possible.
As the temperature is lowered, there is one special temperature at which the distinction between the gas and liquid phases begins to disappear. This temperature is known as the critical temperature, \(T_c\).
The critical point is defined by three critical constants:
| Quantity | Symbol | Description |
|---|---|---|
| Critical Temperature | \(T_c\) | The highest temperature at which a liquid phase can exist. |
| Critical Pressure | \(P_c\) | The pressure required to reach the critical point. |
| Critical Volume | \(V_c\) | The molar volume at the critical point. |
At the critical point, the gas-liquid boundary disappears. Above the critical temperature, no amount of pressure can liquefy the gas because the molecules possess too much kinetic energy for intermolecular attractions to hold them together in a liquid phase.
A substance above both its critical temperature and critical pressure is called a supercritical fluid. Supercritical fluids exhibit properties intermediate between those of liquids and gases and have many important industrial applications.
One advantage of the van der Waals equation is that it predicts the existence of critical behavior. At the critical point, the isotherm contains a horizontal inflection point. This means that both the first and second derivatives of pressure with respect to volume are zero:
\[ \left( \frac{\partial p} {\partial V} \right)_T = 0 \]
\[ \left( \frac{\partial^2 p} {\partial V^2} \right)_T = 0 \]
Together with the van der Waals equation of state, these conditions allow expressions for the critical constants to be derived:
\[ T_c = \frac{8a} {27Rb} \]
\[ P_c = \frac{a} {27b^2} \]
\[ V_c = 3b \]
These relationships connect macroscopic critical behavior to the molecular parameters \(a\) and \(b\), which describe intermolecular attractions and molecular size.
Critical behavior provides one of the clearest demonstrations that real gases differ from ideal gases. Ideal gases can never condense because they possess neither intermolecular attractions nor finite molecular size.
The existence of critical points demonstrates that intermolecular forces play an essential role in determining the behavior of real substances. Measurements of critical temperatures, pressures, and volumes are therefore important sources of information about molecular interactions.
Big picture: Critical behavior marks the transition between ordinary gas-liquid behavior and the supercritical region where the distinction between gas and liquid disappears. The critical constants provide a bridge between measurable thermodynamic properties and the molecular interactions responsible for real-gas behavior.
Real gases differ significantly in their molecular sizes, intermolecular forces, critical temperatures, and critical pressures. At first glance, this suggests that each gas should require its own unique equation of state. Surprisingly, however, many gases exhibit remarkably similar behavior when their properties are expressed relative to their critical constants.
This observation is known as the Principle of Corresponding States. Proposed by Johannes van der Waals, the principle states that substances behave similarly when compared at the same values of appropriately scaled, or reduced, variables.
The reduced variables are defined using the critical constants of the substance:
\[ T_r = \frac{T}{T_c} \]
\[ P_r = \frac{P}{P_c} \]
\[ V_r = \frac{V}{V_c} \]
where \(T_c\), \(P_c\), and \(V_c\) are the critical temperature, pressure, and molar volume of the substance.
Note: Because of how they are defined, reduced variables are always unitless.
Reduced variables provide a convenient way to compare different substances. For example, consider two gases:
Although the temperatures are different, both gases have the same reduced temperature:
\[ T_r = \frac{300\ K}{150\ K} = \frac{600\ K}{300\ K} = 2.0 \]
The Principle of Corresponding States predicts that these gases should exhibit similar thermodynamic behavior because they are at the same reduced temperature.
This allows experimental data from many different substances to be compared on a common scale.
One of the observations that motivated the Principle of Corresponding States is that many substances have similar values of the compression factor at the critical point:
\[ Z_c = \frac{P_cV_c} {RT_c} \]
The van der Waals equation predicts
\[ Z_c = 0.375 \]
irrespective of the substance being considered. Although real gases do not all have exactly this value, many possess critical compression factors that are surprisingly similar.
This suggests that gases near their critical points share common physical behavior despite differences in molecular structure.
The Principle of Corresponding States provides a powerful way to generalize the behavior of real substances. Rather than treating every gas as completely unique, it allows many thermodynamic properties to be expressed in terms of reduced variables that apply broadly across different compounds.
Engineers and physical chemists frequently use reduced variables when estimating thermodynamic properties for gases and fluids whose behavior has not been measured directly.
Big picture: The Principle of Corresponding States reveals that many real gases behave similarly when compared relative to their critical temperatures, pressures, and volumes. By expressing properties in terms of reduced variables, the behavior of seemingly different substances can often be described using a common framework.
Big picture: Real gases deviate from ideal behavior because molecules occupy space and interact with one another. Equations of state such as the van der Waals and Virial equations quantify these deviations, while concepts such as the Boyle Temperature, critical behavior, and the Principle of Corresponding States provide insight into the common patterns that govern the behavior of real substances.