Kinetic Molecular Theory

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1 Kinetic Molecular Theory Particle volume - The volume of an individual gas particle is small compaired to that of its container. Therefore, gas particles are considered to have mass, but no volume. There is a lot of empty space between the gas particles compared to the size of the particles. Gases are highly compressible. Particle motion - Gas particles are in constant straight-line motion, except for when they collied with each other or the sides of the container. Pressure exerted on the sides of the container is the result of the collisions of all the gas particles present. Particle collisions - Collisions between gas particles are perfectly elastic. The total kinetic energy of the particles is constant. The average kinetic energy of the gas particles is directly proportional to the Kelvin temperature.

2 Properties of Gases Expand to completely fill their container Take the shape of their container Low density compared to solids or liquids Compressible Mixtures of gases are always homogeneous Fluid

3 Gas Laws Explained by Kinetic Molecular Theory

4 Boyle's law: A Kinetic Theory View The volume of a gas is inversely proportional to the pressure. Decreasing the volume forces the molecules into a smaller space. Since the velocity of the molecules does not change, more molecules will collide with the container at any one instant, increasing the pressure.

5 Boyle's law: A Kinetic Theory View

6 Charles' law: A Kinetic Theory View The volume of a gas is directly proportional to the absolute temperature. Increasing the temperature increases their average speed, causing them to hit the wall harder and more frequently on average. Since the external pressure remains constant, To keep the internal pressure constant, the volume must increase.

7 Charles' law: A Kinetic Theory View

8 Amonton s Law: A Kinetic Theory View The amount of gas and its volume are the same in either case, but if the gas in the ice bath (O ºC) exerts a pressure of 1 atm, the gas in the boiling-water bath (100 ºC) exerts a pressure of 1.37 atm. The frequency and the force of the molecular collisions with the container walls are greater at the higher temperature.

9 Avogadro s Law -Kinetic Theory View The volume of a gas is directly proportional to the number of gas molecules. Velocity of the molecules does not change. Increasing the number of gas molecules causes more of them to hit the wall at the same time. To keep the pressure constant, the volume must then increase.

10 Dalton s Law -Kinetic Theory View Gas molecules are negligibly small and don t interact. The molecules behave independently of each other, each gas contributing its own collisions to the container with the same average kinetic energy. Because the average kinetic energy is the same, the total pressure is the sum of the pressures of the separate collisions.

11 Kinetic Energy and Molecular Velocities Average kinetic energy of the gas molecules depends on the average mass and velocity. KEave = ½mv 2 Gases in the same container have the same temperature, therefore they have the same average kinetic energy. If they have different masses, the only way for them to have the same kinetic energy is to have different average velocities. Lighter particles will have a faster average velocity than more massive particles.

12 Molecular Speed vs. Molar Mass To have the same average kinetic energy, heavier molecules must have a slower average speed.

13 Temperature and Molecular Velocities KEavg = ½NAmu 2 NA is Avogadro s number KEavg = 1.5RT R is the gas constant in energy units, J/mol K (1 J = 1 kg m 2 /s 2 ) Equating and solving we get 1.5RT = ½NAmu 2 NA mass = molar mass in kg/mol As temperature increases, the average velocity increases

14 Temperature vs. Molecular Speed As the absolute temperature increases, the average velocity increases and the distribution function spreads out, resulting in more molecules with faster speeds.

15 Molecular Velocities All the gas molecules in a sample can travel at different speeds. However, the distribution of speeds follows a statistical pattern called a Boltzman distribution. The method of choice for average velocity is called the root-mean-square method, where the rms average velocity, urms, is the square root of the average of the sum of the squares of all the molecule velocities.

16 Calculate the rms velocity of O2 at 25 C MM, T u rms

17

18 Practice Calculate the rms velocity of CH4 (MM 16.04) at 25 C MM, T u rms

19 Mean Free Path Molecules in a gas travel in straight lines until they collide with another molecule or the container. The average distance a molecule travels between collisions is called the mean free path. Mean free path decreases as the pressure increases.

20 Diffusion and Effusion The process of a collection of molecules spreading out from high concentration to low concentration is called diffusion. The process by which a collection of molecules escapes through a small hole into a vacuum is called effusion. The rates of diffusion and effusion of a gas are both related to its rms average velocity. For gases at the same temperature, this means that the rate of gas movement is inversely proportional to the square root of its molar mass.

21 Diffusion and Effusion Diffusion is the mixing of gas molecules by random motion under conditions where molecular collisions occur. Effusion is the escape of a gas through a pinhole without molecular collisions.

22 Graham s Law of Effusion Thomas Graham ( ) For two different gases at the same temperature, the ratio of their rates of effusion is given by the following equation: The rate of gas movement is inversely proportional to the square root of its molar mass.

23 Calculate the ratio of rate of effusion for oxygen to hydrogen. O2, g/mol; H g/mol =? This means that, on average, the O2 molecules are traveling at ¼ the speed of H2 molecules.

24 Calculate the molar mass of a gas that effuses at a rate times N2. MM =? rate A /rate B, MM N2 MM unknown

25 Ideal vs. 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. PV = nrt n = PV/RT = 1

26 Ideal vs. Real Gases This graph shows how real gas's behavior deviates from ideal behavior as pressure increases. If a gas were to behave perfectly ideally, then the ratio PV/RT would equal exactly 1 for one mole of gas (dashed line).

27 Ideal vs. Real Gases This graph shows how a real gas's behavior deviates from ideal behavior as pressure increases. Each curve represents the behavior of the gas at a different temperature. If a gas were to behave perfectly ideally, then the ratio PV/RT would equal exactly 1 for one mole of gas (dashed line).

28 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.

29 The Effect of Molecular Volume Johannes van der Waals ( ) At high pressure, the amount of space occupied by the molecules is a significant amount of the total volume. 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.

30 Real Gas Behavior Because real molecules attract each other, the molar volume of a real gas is smaller than predicted by the ideal gas law at low temperatures.

31 The Effect of Intermolecular Attractions At low temperature, the attractions between the molecules is significant. The intermolecular attractions makes the real pressure less than the ideal gas law would predict. 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.

32 van der Waals Equation Combining the equations to account for molecular volume and intermolecular attractions we get the following equation used for real gases:

33 Van der Waals Constants for Some Common Gases Van der Waals equation for n moles of a real gas (P + n2 a )(V nb) = nrt 2 V adjusts P up adjusts V down Gas a atm*l 2 mol 2 b L mol He Ne Ar Kr Xe H 2 N 2 O 2 Cl 2 CO 2 CH 4 NH 3 H 2 O

34 Real Gases A plot of PV/RT vs. 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 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 the most important factor is the molecular volume.

35 PV/RT Plots

36 Real Gas Behavior vs Ideal Gas Behavior The volume taken up by the gas particles themselves is less important at lower pressure (a) than at higher pressure (b). As a result, the volume of a real gas at high pressure is somewhat larger than the ideal value.

37 Real Gas Behavior vs Ideal Gas Behavior

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