Real Gases

We've gone in decent detail about how ideal gases behave but in order to bridge the gap between an ideal gas and a real gas, we have to let go of a few assumptions. First off, we assume that the gas particles occupy no space, which is false.

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Since the particles have volume, the true volume the gases can move through is the total volume of the container minus the volume of all the gas particles.

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The particles also attract each other due to their intermolecular forces so the assumption that the particles don't interact outside collisions is also false for real gases. This affects the pressure because the net force from the particles on the walls will change with the introduction of attractive forces between neighboring particles. You would add this correction factor because if the pressure was very low and you subtracted, then you'd get a negative pressure which isn't really a useful quantity. These forces only become more powerful when the distance between particles decreases. The average distance between particles is related to the volume of the container so as the volume increases, the intermolecular attractions decrease. Since these attractions cause particles to clump up, the pressure is lower than that of a real gas.

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Let's mathematically analyze these correction factors starting with volume. Since the volume of the particles themselves is taken into account, the volume of a real gas is less than that of an ideal gas.

The correction factor above makes it clear how we analyze this correction properly. The factor b is a characteristic constant for the gas which is basically how much volume one mole of particles takes up. Since the units are still liters, the units for b are L/mol. For example, the correction factor b for CO2(carbon dioxide) is 0.043 L/mol, which means a mole of carbon dioxide gas in real life takes up about 0.043 L. If we have multiple moles, then we multiply b by n, giving us the second term on the left side of the equation. The more moles of gas, the more volume the gas takes up, which should intuitively make sense.

The correction above is derived using very advanced algebra and intensive geometry but it can be intuitively explained using visual relations. If the volume of the gas is greater, the intermolecular attractions will be weaker because the particles will on average be farther apart. That's why volume is in the denominator. However, the n squared on the top also makes sense because if there's more molecules, then there's more attractions.

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The correction factor a is a measure of the average strength of the intermolecular forces between the particles. It is measured in L² kPa/mol, but other units of pressure like bars can be used.

Let's combine this into the left-hand term of the ideal gas law:

Notice how the pressure term is the ideal pressure and the volume term is the real volume. Why so? Well, the volume term is the "available" volume for the gas particles to travel through. Because of this, we need the real volume because the ideal volume isn't representative of the container. Of course, we want to still compare real gases with this equation to ideal gases given by the rest of the equation so we still use ideal pressure.

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So, if real gases have very prevalent implications, when can we simplify a gas's behavior to that of an ideal gas? At low pressures and high temperatures. At low pressures, the space between particles is much more negligible so the effects of intermolecular forces are diminished. At high temperatures, the particles are moving with a higher average kinetic energy so they'll be farther apart on average, further diminishing intermolecular attractions and neglecting the volumes of the gas particles themselves.

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You can think of it another way. If the gas has a lower pressure and a higher temperature, assuming the moles are constant, the volume will be significantly higher. Since the volume is much greater, the volume of individual particles is far more negligible and the intermolecular forces at play are downplayed heavily due to the average distance between particles being far greater. Even if the moles increased, you'd have to increase them a lot in order to make that increase have an effect.

The following table gives a reference of a handful of gases and their correction factors.