Lecture No. 5
Properties of Gases (5)
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8. Real gases: deviation from the ideal gas law
Real gas vs. ideal gas
• The combined gas law stated that the ratio PV/T is a constant.
• This is not quite true for real gases. For example, if we make a graph of the actual values of PV/T as a function of pressure for a real gas, like O2, we get the following curve:
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8. Real gases: deviation from the ideal gas law
Real gas vs. ideal gas
• A real gas deviates from ideal behavior for two reasons:
First, the model of an ideal gas assumes that gas molecules individually have no volume, but of course they do. We can visualize the sum of all of their individual volumes by supposing that all molecular motions stop and the molecules settle to the bottom of the container.
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8. Real gases: deviation from the ideal gas law
Real gas vs. ideal gas
The measurement of the container’s total volume is the value of V that we use in the gas law calculations. We symbolize it as Vexp. The model of an ideal gas assumes Vexp to be the empty-container volume. But, the space still left for the kinetic motions of the molecules of the real gas (called Vactual is not large as Vexp. The space taken up by the molecules, Vexcluded, would be unavailable to a few extra gas molecules if now added them to the container.
The excluded volume Vexluded is a very tiny fraction of Vexp(i.e., Vexcluded/Vexp). This is why real gases fit the gas law well at ordinary pressures. 4
8. Real gases: deviation from the ideal gas law
Real gas vs. ideal gas
However, if the pressure increased by forcing more molecules into the container, the excluded volume will increase but the value of Vexp stays the same, so the fraction Vexcluded/Vexp increases with the pressure.
Therefore, Vexp become less and less a true description of the actual volume in which gas molecules are free to move.
Therefore, at higher pressures, the volume of the real gas is larger than expected for an ideal gas, so the ratio PV/T is larger than that for the ideal gas.
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8. Real gases: deviation from the ideal gas law
Real gas vs. ideal gas
The second fact is that real gas particles do attract/repel each other, unlike one assumption about the ideal gas.
At lower pressures, these attraction cause smaller values of P than expected from the ideal gas. Thus the ratio PV/T is less than for an ideal gas at the lower pressures.
• Therefore, a correction to the experimental data is need for the real gas.
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8. Real gases: deviation from the ideal gas law
Van der Waals’ equation
• Van der Waal found ways to correct the measured values of P and V to give values that fit the ideal gas law.
• For the excluded volume, it has to be proportional to the number of moles, n, of the gas in the sample, i.e.,
Vexcluded α n Vexcluded = nb where b is the proportionality constant
• The constant b is itself a measure of the excluded volume per mole; larger value of b means larger molecular size.
• The Vexcluded or nb is subtracted from Vexp to give the actual volume (called Videal) remaining for the free motions of the gas molecules, i.e., Videal = Vexp − nb 7
8. Real gases: deviation from the ideal gas law
Van der Waals’ equation
• To correct the pressure term, van der Waals reasoned that the small attractions between gas molecules make the measure pressure of the real gas, Pexp, less than the pressure, Pideal, if the gas were ideal.
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8. Real gases: deviation from the ideal gas law
Van der Waals’ equation
• As shown in the previous figure, the ideal gas particles move in straight lines between collision (a postulate of the kinetic theory). But, if the gas molecules attract each other in a real gas, then when they get close, they must change their directions and move in curved paths as they pass each other. This would make real gas molecules travel longer distances to reach the walls. They would thus take more time and would hit any unit area of the walls less frequently than if the gas were ideal.
• A decreased frequency of collision with the walls in a real gas means a lower pressure.
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8. Real gases: deviation from the ideal gas law
Van der Waals’ equation
• To correct the pressure, van der Waals added a term to the measured pressure. The correction is proportional to the square of the particles concentration, i.e.,
where a is the proportionality constant.
• When this is added to the experimental pressure:
pressure correction a ( n Vexp )2 pressure correction = n2a
Vexp2
Pideal = Pexp + n2a
Vexp2 10
8. Real gases: deviation from the ideal gas law
Van der Waals’ equation
• Constant a corrects for attractions between molecules. a is proportional to the strengths of these attractive forces, large value of a (meaning the stronger the attractive forces) the greater must be the total correction to the measure pressure. We expect this because the stronger the attractive forces are, the greater will be deviations in the paths of the moving molecules. As a result, the collisions with the walls will occur even less frequently. So, with strong attractive forces, implied by large values of a, large corrections to measured pressure are needed.
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8. Real gases: deviation from the ideal gas law
Van der Waals’ equation
• If we now substitute the corrected expressions for the volume and pressure, Videal and Pideal into the equation of state for an ideal gas, we obtain what is called van der Waals equation of state for a real gas.
• The values of P, V and T in the above equation refer to the experimental values. The constants a and b are called the van der Waals constants.
P+ n2a V2 æ
èç ö
ø÷(V -nb)= nRT
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End of Lecture 5
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