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**The Kinetic Molecular Theory**

explains the behavior of gases at the molecular level Questions Concerning Gas Behavior 1. Origin of Pressure: Pressure is a measure of the force a gas exerts on a surface. How do individual gas particles create this force? 2. Boyles Law: A change in gas pressure in one direction causes a change in gas volume in the other. What happens to the particles when external pressure compresses the gas volume? Why aren’t liquids and solids compressible? 3. Dalton’s Law: The pressure of a gas mixture is the sum of the pressures of the individual gases. Why does each gas contribute to the total pressure in proportion to its mole fraction? 4. Charles Law: A change in temperature is accompanied by a corresponding change in volume. What effect does higher temperature have on gas particles that increase the volume-or increases the pressure if volume is fixed? This raises a more fundamental question: what does temperature measure on the molecular scale? 5. Avagadro’s Law: Gas volume (or pressure) depends on the number of moles present, not on the nature of the particular gas. But shouldn’t one mole of larger molecules occupy more space than one mole of smaller ones? And why doesn’t’ one mole of heavier molecules exert more pressure than one mole of lighter ones?

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**Postulates of the Kinetic Molecular Theory**

Postulate 1. Particle Volume. A gas consists of a large collection of individual particles. The volume of an individual particle is extremely small compared to the volume of the container. Most of the space occupied by a gas is empty. Postulate 2. Particle Motion. Gas particles are in constant motion. Their motion is random and in straight lines. At any one time there are gas particles moving in all directions. When they collide with the walls of the container or one another they change their direction. Postulate 3. Particle Collisions. The forces of attraction between molecules in a gas are negligible so that they do not influence one another unless they collide. When they collide they can transfer kinetic energy.

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**Postulate 4. Temperature**

Postulate 4. Temperature. In a collection of molecules at a certain temperature the total energy remains constant although each individual molecule may have a certain kinetic energy and can transfer energy in a collision. Maxwell distribution of molecular speeds molecules have a distribution of speeds. they could be moving very fast one second, collide with another molecule and transfer energy to another molecule and be sitting almost still at another instant. the average kinetic energy Ek is proportional to temperature or, Ek = c x T where c is a constant which is the same for any gas.

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**The molecular view of the gas laws:**

1. Origin of Pressure. When a moving object collides with any surface it exerts a force and therefore pressure. From postulate 2 which describes molecular motion, when a particle collides with the container wall, it exerts a force. All of the collisions occurring at any instant make up the observed pressure. The more particles there are the more frequent are the collisions with the wall and therefore the greater the pressure. 2. Boyle’s Law. A gas is composed of very tiny particles of negligible volume and consists of mostly empty space. When an external pressure is applied to the gas, the molecules get closer together, decreasing the volume of the sample. The pressure exerted by the gas increases simultaneously because a smaller volume has shorter distances between gas molecules and the walls so there are more frequent collisions. Solids and liquids are different in that the molecules are already very close together and cannot be compressed.

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3. Dalton’s Law. Adding another component to a gas simply increases the number of particles which increases the frequency of collisions and therefore the pressure. Each gas exerts a fraction of the total pressure based on the fraction of molecules (or moles) of that gas in the mixture.

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4. Charles Law. As the temperature increases, the most probable molecular speed and average kinetic energy increases. Therefore the molecules hit the container walls more frequently and more energetically which causes a greater gas pressure. If we have an apparatus like that below where there is a movable piston, the gas pressure exerts a force on the piston and it moves upward to return the pressure to atmosphere (the external pressure).

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5. Avagadro’s Law. Adding more molecules to a container increases the total number of collisions with the walls and, therefore, the internal pressure. As a result, the volume expands to yield the same pressure as before.

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**We have not explained why equal numbers of molecules of different gases occupy the same volume.**

First we must realize and understand why heavier molecules (O2) do not hit the wall with more energy than lighter molecules (H2) if a light object and a heavy object have the same kinetic energy then the heavy object must have a smaller velocity the average kinetic energy is where is root-mean square speed. A molecule moving with this speed has the average kinetic energy. The rms speed is slightly higher than the most probable speed but they are proportional. Consider molecules traveling in the x direction only ux during an interval Dt the average number of collisions with the right hand wall is, time interval half moving left, half miving right number density x component of velocity area of wall

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each molecule that hits the wall changes its motion from mux to –mux, a total change of 2mux, therefore, the total momentum change in the time interval is, MM total momentum change Newton’s 2nd law states that the force is the rate of change of momentum which is the total momentum change divided by Dt, force is proportional to MM The pressure is the Force per unit Area, Not all the molecules travel with the same velocity so the detected pressure is the average of all molecules A

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The molecules are moving randomly and there is no net flow in a particular direction. Therefore the average speed in the x direction is the same as in the y and z direction, u. or A Subbing this result into equation above.

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**This says that the higher the molar mass, the lower the root-mean-speed.**

In other words, at the same temperature, O2 molecules move more slowly, on average, than H2 molecules but, previously we said that the more massive molecules provide more force.

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**Remember the average kinetic energy,**

substituting in the expression we just derived for urms we get, The average kinetic energy for a large number of molecules is independent of the mass of the molecules and only depends on the temperature. The lighter molecules collide more frequently with the container walls but with less force than the heavier molecules. Both light and heavy (and all in between) molecules create the same pressure and therefore have the same volume.

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eg. What is the rms speed of carbon dioxide molecules in a container of gas at 23 oC and at what temperature does H2 have the same rms speed? Solution.

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Effusion the process by which a gas escapes from its container through a tiny hole into an evacuated space. it makes sense that the rate of effusion is proportional to the root-mean square speed so at a constant temperature the rate of effusion is inversely proportional to the square root of the molar mass, - Grahams law of effusion Ar is lighter than Kr and so effuses faster assuming equal pressures of the two gases. The ratio of the two rates is, At a given temperature and pressure, the gas with the lower molar mass effuses faster because the most probable speed of its molecules is higher, therefore more molecules escape through the tiny hole per unit time.

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**eg. Calculate the ratio of the effusion rates of helium and methane.**

Solution.

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Diffusion diffusion is the movement of one gas through another gas and is actually closely related to effusion. we just calculated that the root mean speed of CO2 is 410 m s-1 (1480 km h-1) at room temperature. CS2 has a urms of, motion of gas molecules under the same conditions. Why then, if I take the top of a container of CS2, do you not smell it immediately? in fact, convection is mainly responsible for the movement of gases at ambient conditions.

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d traveled by NH3 d traveled by HCl NH3(aq) NH4Cl(s) HCl(aq) glass tube

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**Mean-Free Path and Collision Frequency**

Distribution of speeds At room T, N2 travels with an average speed of 0.47 km/s (urms = 0.51 km/s). At any instant one N2 could collide with another molecule and be traveling at 2 km/s and another could be standing still. These extreme speeds are less likely than the average. The mean free path is the average distance traveled between collisions. At ambient pressures an N2 molecule (~4 x m in diameter) travels about 7 x 10-8 m between collisions. To put this into perspective, if a molecule were the size of a tennis ball (10 cm in diameter), it would have to travel about 18 m before hitting another tennis ball.

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The collision frequency is the average number or collisions in a second. We saw that the average distance a molecule travels between collisions is quite far compared to the size of a molecule. If the N2 molecule travels an average of 4.7 x 102 m s-1 and the average distance traveled between collisions is 7 x 10-8 m, the collision frequency is, 7 billion collisions per second!!! - you can see why molecular diffusion through the atmosphere can be so slow! kinetics???? (Ch. 14 – CHEM1051)

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**Real Gases: Deviations From Ideal Behavior**

typically, simple models are much more useful than complex ones as long as they can be used to explain and predict properties. the kinetic molecular theory is quite useful, however, we must realize that in reality molecules have non-zero volumes and there are, in fact, forces of attraction between atoms and molecules in the gas-phase (intermolecular forces which must be present because they cause gases to condense to liquids and/or solids). so, we would expect these real gases to deviate from ideal behavior especially under conditions where theei real properties can be most effective, ie. when the molecules are close together or at low temperature and/or high pressure.

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**Lets examine the two characteristics of real gases which cause deviation from ideal behavior.**

1. Molecular volume at very high external pressures, the free volume is significantly less than the container volume because of the volume of the molecules themselves. at ordinary pressures the free volume is essentially equal to the container volume because the molecules occupy only a tiny fraction of the available space. The effect of molecular volume is to make the volume occupied by a mole of gas larger than predicted by the ideal gas law, PV/RT > 1

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**2. Intermolecular forces.**

The effect of attractive forces between molecules is to decrease the pressure that a mole of gas (at standard temperature in a 22.4 L container) from that which it would exert if it were ideal. PV/RT < 1

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**Effect of extreme pressure,**

for 1 mol of ideal gas, PV/RT is approximately 1.

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H2 and He have extremely weak intermolecular attractions and therefore the PV/RT vs Pext plot, does not exhibit the normal dip that other gases do at moderate pressures. The molecular volume effect predominates at all pressures.

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**Real Gases: The van der Waals Equation**

To describe the behavior of real gases we need to make some corrections to the ideal gas equation which we have already developed. we need to take into account intermolecular forces to adjust the measured pressure up and we need to take into account the molecular volume to adjust the measured volume down. adjusts P up adjusts V down

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**Consider a 1. 98-L vessel containing 215 g (4. 89 mol) of dry ice**

Consider a 1.98-L vessel containing 215 g (4.89 mol) of dry ice. After standing at 299 K, the solid CO2 sublimes to the gas phase. The pressure is measured (Preal = 44.8 atm) and calculated by the ideal gas law (Pideal). Using the van der Waals equation to calculate the pressure, only 2.5 % error vs 35.3 % using the IGL

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**eg. Calculate the molar volume of chlorine at 1 atm and 10 atm at 0 oC**

eg. Calculate the molar volume of chlorine at 1 atm and 10 atm at 0 oC. Compare the values given by the ideal gas equation. Using the ideal gas law, V1atm = 22.4 L and V10atm = 2.24 L Now, for a real gas, can’t solve for V? Sure we can, there are two ways… 1. Assume n2a/V2 is small compared with P and substitute V = nRT/P

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**2. We can also employ an “iterative method” using a spread sheet like excel…**

V (vdw) 2240 L 224 L 22.2 L 7.24 L 4.26 L 2.04 L 0.154 L assumption V (exact vdw ) 2240 L 224 L 22.2 L 7.23 L 4.23 L 1.98 L L iterative method P / atm 0.01 0.1 1 3 5 10 100 V (ideal gas law) 2240 L 224 L 22.4 L 7.47 L 4.48 L 2.24 L 0.224 L Temperature?

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5.7/5.1 Kinetic Molecular Theory of Gases

5.7/5.1 Kinetic Molecular Theory of Gases

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