1. Pressure and KMT mvx−m(−vx)=2mvx
The macroscopic phenomena of pressure can be explained in terms of the kinetic molecular theory of gases. Assume the case in which a gas molecule (represented by a sphere) is in a box, length L.
Considering the sphere is only moving in the x-direction, we can examine the instance of the sphere colliding elastically with one of the walls of the box.
The momentum of this collision is given by p=mv, in this case p=mvx, since we are only considering the x dimension.
The total momentum change for this collision is then given by 
Figure 1
mvx−m(−vx)=2mvx
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2. Pressure and KMT 𝑃= 𝐹 𝐴 𝑃= 𝑚𝑣 𝑥 2 𝐿 3
Given that the amount of time it takes between collisions of the molecule with the wall is
 L/vx
We can give the frequency of collisions of the molecule against a given wall of the box per unit time as vx/2L. One can now solve for the change in momentum per unit of time:
(2mvx)(vx/2L)= mv2x/L
Solving for momentum per unit of time gives the force exerted by an object (F=ma=p/time). 
With the expression that F=mvx2/L one can now solve for the pressure exerted by the molecular collision, where area is given as the area of one wall of the box, A=L2
𝑃= 𝐹 𝐴
𝑃= 𝑚𝑣 𝑥 2 𝐿 3
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3. Pressure and KMT 𝑃= 𝑁𝑚𝑣 𝑥 2 𝑉 𝑃= 𝑁𝑚 𝑣 2 3𝑉
The expression can now be written in terms of the pressure associated with collisions from N number of molecules:
𝑃= 𝑁𝑚𝑣 𝑥 2 𝑉
This expression can now be adjusted to account for movement in the x, y and z directions by using mean-square velocity for three dimensions and a large value of N. The expression now is written as:
𝑃= 𝑁𝑚 𝑣 2 3𝑉
This expression now gives pressure, a macroscopic quality, in terms of atomic motion. The significance of the above relationship is that pressure is proportional to the mean-square velocity of molecules in a given container. Therefore, as molecular velocity increases so does the pressure exerted on the container.
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4. Real Gases
Real gases often do not behave like ideal gases at high pressure or low temperature.
Ideal gas laws assume
1. no attractions between gas molecules.
2. gas molecules do not take up space.
Based on the kinetic molecular theory
At low temperatures and high pressures these assumptions are not valid.
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5. The Effect of the Finite Volume of Gas Particles
At low pressures, the molar volume of argon is nearly identical to that of an ideal gas.
But as the pressure increases, the molar volume of argon becomes greater than that of an ideal gas.
At the higher pressures, the argon atoms themselves occupy a significant portion of the gas volume, making the actual volume greater than that predicted by the ideal gas law.
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6. Real Gas Behavior
Because real molecules take up space, the molar volume of a real gas is larger than predicted by the ideal gas law at high pressures.
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7. Modification of the Ideal Gas Equation
In 1873, Johannes van der Waals (1837–1923) modified the ideal gas equation to fit the behavior of real gases at high pressure.
The molecular volume makes the real volume larger than the ideal gas law would predict.
Van der Waals modified the ideal gas equation to account for the molecular volume.
b is called a van der Waals constant and is different for every gas because their molecules are different sizes.
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8. The Effect of Intermolecular Attractions
At high temperature, the pressures of the gases are nearly identical to that of an ideal gas.
But at lower temperatures, the pressures of the gases are less than that of an ideal gas.
At the lower temperatures, the gas atoms spend more time interacting with each other and less time colliding with the walls, making the actual pressure less than that predicted by the ideal gas law.
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9. The Effect of Intermolecular Attractions
Van der Waals modified the ideal gas equation to account for the intermolecular attractions.
a is another van der Waals constant and is different for every gas because their molecules have different strengths of attraction.
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10. Van der Waals’s Equation
Combining the equations to account for molecular volume and intermolecular attractions we get the following equation.
Used for real gases
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11. Real Gases
A plot of PV/RT versus P for 1 mole of a gas shows the difference between real and ideal gases.
It reveals a curve that shows the PV/RT ratio for a real gas is generally lower than ideal for “low” pressures—meaning that the most important factor is the intermolecular attractions.
It reveals a curve that shows the PV/RT ratio for a real gas is generally higher than ideal for “high” pressures—meaning that the most important factor is the molecular volume.
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12. Real versus Ideal Behavior
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13. Van der Waals’s Equation
If SO2 were an ideal gas, the pressure at 0.0°C exerted by mole occupying L would be atm. Use the van der Waals equation to estimate the pressure of this volume of mol SO2 at 0.0°C.
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14. Van der Waals’s Equation
Rearranging the van der Waal’s equation to give P
Substituting values of R,T, V, a, and b
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©2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.