10.7 Kinetic Molecular Theory Kinetic Molecular Theory. Kinetic Molecular Theory. Kinetic Molecular Theory. Kinetic Molecular Theory
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1 The first scheduled quiz will be given next Tuesday during Lecture. It will last 5 minutes. Bring pencil, calculator, and your book. The coverage will be pp , i.e. Sections 0.0 through Theory developed to explain gas behavior. Theory based on properties at the molecular level. Kinetic molecular theory gives us a model for understanding pressure and temperature at the molecular level. Pressure of a gas results from the number of collisions per unit time on the walls of container. 0.7 There is a spread of individual energies of gas molecules in any sample of gas. As the temperature increases, the average kinetic energy of the gas molecules increases. Assumptions: Gases consist of a large number of molecules in constant random motion. Volume of individual molecules negligible compared to volume of container. Intermolecular forces (forces between gas molecules) negligible. Energy can be transferred between molecules, but total kinetic energy is constant at constant temperature. Average kinetic energy of molecules is proportional to temperature. Magnitude of pressure given by how often and how hard the molecules strike. Gas molecules have an average kinetic energy. Each molecule may have a different energy. As kinetic energy increases, the velocity of the gas molecules increases. Root mean square speed, u, is the speed of a gas molecule having average kinetic energy. Average kinetic energy, ε, is related to root mean square speed: ε = mu
2 Do you remember how to calculate v xy from v x and v y? xy ( v v ) v = + x y And how about v from all three components? [ v + v v ] v = + x y z Remember these equations!! They ll pop up again in Chap.. Most Probale Speed = v mp RT = M u mp <u> u rms Average Speed = 8RT v = π M rms Speed = v rms 3RT = M 8 And, v mp : v : v rms = : : 3 π = :.8 :.5. Be careful of speed versus velocity. The former is the magnitude of the latter.. The momentum of a molecule is p = mv. During a collision, the change of momentum is p wall = p final p initial = (-mv x ) (mv x ) = mv x. Now we have PV = N mu and PV = nrt 3 But N = nn 0, so we can divide both sides by n to obtain 3 Nmu 0 = RT, butnm 0 = M, so 3 Mu = RT 3. t = l / v x p x / t =... = mv x / l, where l is length of the box 4. force = f = ma = m( v / t) = p / t = mv x / l = force along x 5. And for N molecules, F = N(m(v x ) avg / l ) ( ) 6. But ( v ) = v = N v + v + v v 3 x avg x x x x xn 7. And F Nm P = = vx A Al and Al = V so that PV = Nm v x u = v + v + v = 3 v so that PV = Nmu x y z x 3
3 Application to Gas Laws As volume increases at constant temperature, the average kinetic of the gas remains constant. Therefore, u is constant. However, volume increases so the gas molecules have to travel further to hit the walls of the container. Therefore, pressure decreases. If temperature increases at constant volume, the average kinetic energy of the gas molecules increases. Therefore, there are more collisions with the container walls and the pressure increases. Molecular Effusion and Diffusion As kinetic energy increases, the velocity of the gas molecules increases. Average kinetic energy of a gas is related to its mass: ε = mu Consider two gases at the same temperature: the lighter gas has a higher rms than the heavier gas. Mathematically: u = 3RT M Molecular Effusion and Diffusion The lower the molar mass, M, the higher the rms. Graham s Law of Effusion As kinetic energy increases, the velocity of the gas molecules increases. Effusion is the escape of a gas through a tiny hole (a balloon will deflate over time due to effusion). The rate of effusion can be quantified. Graham s Law of Effusion Consider two gases with molar masses M and M, the relative rate of effusion is given by: r M = r M Only those molecules that hit the small hole will escape through it. Therefore, the higher the rms the more likelihood of a gas molecule hitting the hole. 3
4 Graham s Law of Effusion Consider two gases with molar masses M and M, the relative rate of effusion is given by: 3RT r u M M = = = r u 3RT M M Only those molecules that hit the small hole will escape through it. Therefore, the higher the rms the more likelihood of a gas molecule hitting the hole. Diffusion and Mean Free Path Diffusion of a gas is the spread of the gas through space. Diffusion is faster for light gas molecules. Diffusion is significantly slower than rms speed (consider someone opening a perfume bottle: it takes while to detect the odor but rms speed at 5 C is about 50 mi/hr). Diffusion is slowed by gas molecules colliding with each other. Diffusion and Mean Free Path Average distance of a gas molecule between collisions is called mean free path. At sea level, mean free path is about cm. Real Gases: Deviations From the ideal gas equation, we have PV = n RT or PV nrt For mol of gas, PV/nRT = for all pressures. In a real gas, PV/nRT varies from significantly and is called Z. Z = PV nrt The higher the pressure the more the deviation from ideal behavior. = Real Gases: Deviations From the ideal gas equation, we have PV = n RT For mol of gas, PV/RT = for all temperatures. As temperature increases, the gases behave more ideally. The assumptions in kinetic molecular theory show where ideal gas behavior breaks down: the molecules of a gas have finite volume; molecules of a gas do attract each other. 4
5 Real Gases: Deviations As the pressure on a gas increases, the molecules are forced closer together. As the molecules get closer together, the volume of the container gets smaller. The smaller the container, the more space the gas molecules begin to occupy. Therefore, the higher the pressure, the less the gas resembles an ideal gas. Real Gases: Deviations As the gas molecules get closer together, the smaller the intermolecular distance. Real Gases: Deviations The smaller the distance between gas molecules, the more likely attractive forces will develop between the molecules. Therefore, the less the gas resembles and ideal gas. As temperature increases, the gas molecules move faster and further apart. Also, higher temperatures mean more energy available to break intermolecular forces. Real Gases: Deviations Therefore, the higher the temperature, the more ideal the gas. The first scheduled quiz will be given next Tuesday during Lecture. It will last 5 minutes. Bring pencil, calculator, and your book. The coverage will be pp , i.e. Sections 0.0 through.4. 5
6 Real Gases: Deviations The van der Waals Equation We add two terms to the ideal gas equation one to correct for volume of molecules and the other to correct for intermolecular attractions The correction terms generate the van der Waals equation: nrt n a P = V nb V where a and b are empirical constants characteristic of each gas. Corrects for molecular volume Real Gases: Deviations The van der Waals Equation nrt n a P = V nb V General form of the van der Waals equation: n a P + ( V nb) = nrt V Corrects for molecular attraction Chapter -- Intermolecular Forces, Liquids, and Solids In many ways, this chapter is simply a continuation of our earlier discussion of real gases. This plot for SO is a more representative one of real systems!!! Remember this nice, regular behavior described by the ideal gas equation. 6
7 And this is a plot for an ideal gas of the dependence of Volume on Temperature. Now this one includes a realistic one for Volume as a function of Temperature! Why do the boiling points vary? Is there anything systematic? London Dispersion Forces Hydrogen Bonding Dipole-Dipole Forces 7
8 Intermolecular Forces -- forces between molecules -- are now going to be considered. Note that earlier chapters concentrated on Intramolecular Forces, those within the molecule. Important ones: ion-ion ion-dipole dipole-dipole dipole-induced dipole similar to atomic systems (review definition of dipoles) London Dispersion Forces: induced dipole-induced dipole polarizability Hydrogen Bonding How do you know the relative strengths of each? Virtually impossible experimentally!!! Most important though: Establish which are present. London Dispersion Forces: Always All others depend on defining property such as existing dipole for d-d. It has been possible to calculate the relative strengths in a few cases. Relative Energies of Various Interactions d-d d-id disp Ar N C 6 H C 3 H HCl 6 06 CH Cl SO H O HCN Let s take a closer look at these interactions: Ion-dipole interaction 8
9 Let s take a closer look at dipole-dipole interactions. This is the simple one. But we also have to consider other shapes. Review hybridization and molecular shapes. Recall the discussion of sp, sp, and sp 3 hybridization? A Polarized He atom with an induced dipole molecule F Cl Br I CH 4 polarizability molecular wt Molecular Weight predicts the trends in the boiling points of atoms or molecules without dipole moments because polarizability tends to increase with increasing mass. London dispersion forces (interactions) 9
10 Water provides our best example of Hydrogen Bonding. But polarizability also depends on shape, as well as MW. But hydrogen bonding is not limited to water: These boiling points demonstrate the enormous contribution of hydrogen bonding. Water is also unusual in the relative densities of the liquid and solid phases. The crystal structure suggests a reason for the unusual high density of ice. 0
11 But water isn t the only substance to show hydrogen bonding!.3 Some Properties of Liquids Examples of Viscosity Viscosity the resistance to flow of a liquid, such as oil, water, gasoline, molasses, (glass!!!) Surface Tension tendency to minimize the surface area compare water, mercury Cohesive forces bind similar molecules together Adhesive forces bind a substance to a surface Capillary action results when these two are not equal Soap reduces the surface tension, permitting one material to wet another more easily The unit of viscosity is poise, which is g/cm-s, but typical values are much smaller and are usually listed as cp = 0.0 P. Rationale for Surface Tension Surface Tension Surface molecules are only attracted inwards towards the bulk molecules. Therefore, surface molecules are packed more closely than bulk molecules. Surface tension is the amount of energy required to increase the surface area of a liquid, in J/m. Cohesive forces bind molecules to each other. Adhesive forces bind molecules to a surface.
12 Surface Tension Meniscus is the shape of the liquid surface. If adhesive forces are greater than cohesive forces, the liquid surface is attracted to its container more than the bulk molecules. Therefore, the meniscus is U-shaped (e.g. water in glass). If cohesive forces are greater than adhesive forces, the meniscus is curved downwards. Capillary Action: When a narrow glass tube is placed in water, the meniscus pulls the water up the tube. Remember that surface molecules are only attracted inwards towards the bulk molecules. also called FUSION Phase Changes H vap : 40,670 J/mol Sublimation: solid gas. Vaporization: liquid gas. Melting or fusion: solid liquid. Deposition: gas solid. Condensation: gas liquid. Freezing: liquid solid. H fus : 6,00 J/mol C p (s): 37.6 J/mol-K C p (l): 7.4 J/mol-K C p (g): 33. J/mol-K
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