Name: real gases: when the ideal gas law breaks down
Uploaded: 2020-09-03T13:38:27.000Z
Duration: 4 min 1 s
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Real Gases: Effects of Intermolecular Forces and Molecular Volume Deriving Van der Waals Equation

Ideal gases follow the relation PV over nRT equals one. Recall Boyle&#8217;s law, which states that when the amount and temperature of a gas are held constant, increasing the pressure will invariably decrease the volume to maintain a constant ratio. 

But when that ratio is plotted as a function of pressure for one mole of several real gases, it equals the ideal value &#8220;one&#8221; only at low pressures. As the pressure increases, the curves deviate significantly from ideality. 

At low pressures, the combined volume of gas particles is negligible relative to the container volume &#8212; like a pea inside a basketball. Therefore, the volume available to ideal gas particles equals the total container volume.

At higher pressures, the gas density is much greater. Thus, the combined volume of the gas particles becomes significant &#8212; like a pea inside a ping-pong ball. 

Therefore, the assumption of the kinetic molecular theory that gas particles occupy negligible volume is invalid at high pressures.

The volume occupied by a real gas is greater than the volume available to its particles, which is the volume it would occupy in an ideal case, by nb, where b is an experimentally determined constant that depends on t

real gases: effects of intermolecular forces and molecular volume deriving van der waals equation

Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. However, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. 

According to kinetic molecular theory, particles of an ideal gas do not exhibit attractive or repulsive forces on one another. They are assumed to have a negligible volume compared with that of the container. At room temperature and 1 atm or less, gases follow the ideal behavior, as implied by the ideal gas equation.

<img alt="Ideal gas law equation \(PV=nRT\) formula representation." src="/files/core/CDNSource/lessons/1045/1045_Equations-01.png" style="height: 40px;" data-altupdatedat="1761830439000" data-alt="Eq1" />

At higher pressures or lower temperatures, however, deviations from the ideal gas law occur, meaning that the observed relationships between its pressure, volume, and temperature are not accurately followed.

Rearranging the ideal gas equation to solve for n gives:

<img alt="Ideal gas law equation \( n = \frac{PV}{RT} \) formula for chemistry calculations." src="/files/core/CDNSource/lessons/1045/1045_Equations-02.png" style="height: 40px;" data-altupdatedat="1761830439000" data-alt="Eq2" />

For 1 mole of an ideal gas, the ratio PV/RT = 1, regardless of pressure. Any deviation of this ratio from one is an indication of non-ideal behavior.

The ideal gas law does not describe gas behavior well at relatively high pressures. Meaning, the ratio equals 1 only at low pressures. But as the pressure rises, PV/RT begins to deviate from 1, and the deviations are not uniform. At high pressures, the deviation from ideal behavior is large and different for each gas. Real gases, in other words, do not behave ideally at high pressure. At lower pressures (usually below 10 atm), however, the deviation from ideal behavior is small, and we can use the ideal-gas equation.

Particles of a hypothetical ideal gas have no significant volume and do not attract or repel each other. In general, real gases approximate this behavior at relatively low pressures and high temperatures. However, at high pressures, the molecules of a gas are crowded closer together, and the amount of empty space between the molecules is reduced. At these higher pressures, the volume of the gas molecules themselves becomes appreciable relative to the total volume occupied by the gas. The gas, therefore, becomes less compressible at these high pressures, and although its volume continues to decrease with increasing pressure, this decrease is not proportional as predicted by Boyle&#8217;s law.

At relatively low pressures, gas molecules have practically no attraction for one another because they are (on average) so far apart, and they behave almost like particles of an ideal gas. At higher pressures, however, the force of attraction is also no longer insignificant. This force pulls the molecules a little closer together, slightly decreasing the pressure (if the volume is constant) or decreasing the volume (at constant pressure). This change is more pronounced at low temperatures because the molecules have lower KE relative to the attractive forces, and so they are less effective in overcoming these attractions after colliding with one another.

There are several different equations that better approximate gas behavior than does the ideal gas law. The first, and simplest of these, was developed by the Dutch scientist Johannes van der Waals in 1879. The van der Waals equation improves upon the ideal gas law by adding two terms: one to account for the volume of the gas molecules and another for the attractive forces between them.

<img alt="Van der Waals gas equation, \( \left[ P + \frac{an^2}{V^2} \right](V-nb) = nRT \), formula graphic." src="/files/core/CDNSource/lessons/1045/1045_Equations-03.png" style="height: 40px;" data-altupdatedat="1761830439000" data-alt="Eq3" />

The constant a corresponds to the strength of the attraction between molecules of a particular gas, and the constant b corresponds to the size of the molecules of a particular gas. The &#8220;correction&#8221; to the pressure term in the ideal gas law is an2/V2, and the &#8220;correction&#8221; to the volume is nb. Note that when V is relatively large and n is relatively small, both of these correction terms become negligible, and the van der Waals equation reduces to the ideal gas law, PV = nRT. Such a condition corresponds to a gas in which a relatively low number of molecules is occupying a relatively large volume, that is, a gas at relatively low pressure.

At low pressures, the correction for intermolecular attraction, a, is more important than the one for molecular volume, b. At high pressures and small volumes, the correction for the volume of the molecules becomes important because the molecules themselves are incompressible and constitute an appreciable fraction of the total volume. At some intermediate pressure, the two corrections have opposing influences, and the gas appears to follow the relationship given by PV = nRT over a small range of pressures.

Strictly speaking, the ideal gas equation functions well when intermolecular attractions between gas molecules are negligible, and the gas molecules themselves do not occupy an appreciable part of the whole volume. These criteria are satisfied under conditions of low pressure and high temperature. Under such conditions, the gas is said to behave ideally, and deviations from the gas laws are small enough that they may be disregarded &#8212; this is, however, very often not the case.

This text is adapted from <a href="https://openstax.org/books/chemistry-2e/pages/9-6-non-ideal-gas-behavior">Openstax, Chemistry 2e, Section 9.2: Non-ideal Gas Behavior.</a>

Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and ...

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real gases: when the ideal gas law breaks down

Real gases are gases that do not always follow the ideal gas law, PV = nRT, especially when pressure is high or temperature is low. The ideal gas model assumes gas particles have no attractive or repulsive forces and take up negligible space compared with the container. At room temperature and 1 atm or less, many gases behave close to ideally. When conditions change, the measured relationships between pressure, volume, and temperature can drift away from the ideal prediction.

One way to see this difference is to examine the ratio PV/RT for 1 mole of gas. For an ideal gas, this ratio equals 1 at any pressure. A change away from 1 signals non-ideal behavior. The ideal gas law works best at low pressures. As pressure rises, the deviation becomes larger, and the size of the deviation is not the same for every gas.

Pressure also affects how closely gas particles are packed. At high pressure, the molecules are crowded together, so the empty space between them becomes smaller. Their own volume then matters more, so the gas is less compressible than Boyle’s law predicts. The volume still decreases as pressure increases, but not in the proportional way expected for an ideal gas.

At low pressure, gas molecules are usually far apart, so attraction between them is very small. At higher pressure, those attractive forces become more important. They pull molecules slightly closer together and can lower the pressure at constant volume or reduce the volume at constant pressure. This effect is stronger at low temperature because the molecules have less kinetic energy and are less able to overcome attraction during collisions.

A better model for real gas behavior is the van der Waals equation, first developed by Johannes van der Waals in 1879. It improves the ideal gas law by adding one correction for molecular attraction and another for molecular volume. The constant a describes the strength of attraction between molecules of a given gas. The constant b describes the size of the molecules.

In the pressure correction, the term a n²/V² accounts for intermolecular attractions. In the volume correction, the term nb accounts for the space taken up by the molecules themselves. When volume is large and the amount of gas is small, both corrections become very small, and the van der Waals equation reduces to PV = nRT. That situation matches a gas at relatively low pressure.

At low pressure, the attraction correction is usually more important than the volume correction. At high pressure and small volume, the volume term becomes more important because gas molecules are not compressible and take up a noticeable part of the total volume. At some intermediate pressure, the two corrections can oppose each other, so the gas may appear to follow the ideal gas law over a limited range. The ideal gas equation is therefore most reliable when intermolecular attractions are negligible and the molecules do not occupy much of the total volume, which is usually the case at low pressure and high temperature.

Real Gases: When the Ideal Gas Law Breaks Down

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Explore how pressure, temperature, molecular attraction, and molecular volume cause real gases to deviate from the ideal gas law and how van der Waals corrects it.

Real Gases: When the Ideal Gas Law Breaks Down | JoVE K12

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Real Gases: When the Ideal Gas Law Breaks Down

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