General Chemistry Chapter 9 Section 5: The Kinetic-Molecular Theory

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I: Recalling that gas pressure is exerted by rapidly moving gas molecules and depends directly on the number of molecules hitting a unit area of the wall per unit of time, we see that the KMT conceptually explains the behavior of a gas as follows:

1.) Amontons's law. If the temperature is increased, the average speed and kinetic energy of the gas molecules increase. If the volume is held constant, the increased speed of the gas molecules results in more frequent and more forceful collisions with the walls of the container, therefore increasing the pressure 2.) Charles's law. If the temperature of a gas is increased, a constant pressure may be maintained only if the volume occupied by the gas increases. This will result in greater average distances traveled by the molecules to reach the container walls, as well as increased wall surface area. These conditions will decrease the both the frequency of molecule-wall collisions and the number of collisions per unit area, the combined effects of which balance the effect of increased collision forces due to the greater kinetic energy at the higher temperature 3.) Boyle's law. If the gas volume is decreased, the container wall area decreases and the molecule-wall collision frequency increases, both of which increase the pressure exerted by the gas 4.) Avogadro's law. At constant pressure and temperature, the frequency and force of molecule-wall collisions are constant. Under such conditions, increasing the number of gaseous molecules will require a proportional increase in the container volume in order to yield a decrease in the number of collisions per unit area to compensate for the increased frequency of collisions 5.) Dalton's Law. Because of the large distances between them, the molecules of one gas in a mixture bombard the container walls with the same frequency whether other gases are present or not, and the total pressure of a gas mixture equals the sum of the (partial) pressures of the individual gases

Kinetic Molecular Theory is based on the following five postulates described here (Note: The term "molecule" will be used to refer to the individual chemical species that compose the gas, although some gases are composed of atomic species, for example, the noble gases.)

1.) Gases are composed of molecules that are in continuous motion, travelling in straight lines and changing direction only when they collide with other molecules or with the walls of a container 2.) The molecules composing the gas are negligibly small compared to the distances between them 3.) The pressure exerted by a gas in a container results from collisions between the gas molecules and the container walls 4.) Gas molecules exert no attractive or repulsive forces on each other or the container walls; therefore, their collisions are elastic (do not involve a loss of energy). 5.) The average kinetic energy of the gas molecules is proportional to the kelvin temperature of the gas

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part II: According to ___________, the molecules of a gas are in rapid motion and the molecules themselves are small. The average distance between the molecules of a gas is large compared to the size of the molecules. As a consequence, gas molecules can move past each other easily and diffuse at relatively fast rates

Graham's law

Molecular Velocities and Kinetic Energy: The postulates of the KMT theory may be applied in a more quantitative fashion to derive these individual laws. To do this, we must first look at velocities and kinetic energies of gas molecules, and the temperature of a gas sample

In a gas sample, individual molecules have widely varying speeds; however, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are constant. This molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed

The kinetic energy (KE) of a particle of mass ( m) and speed ( u) is given by:

KE= 1/2 𝑚𝑢2 Expressing mass in kilograms and speed in meters per second will yield energy values in units of joules (J = kg m2 s-2). To deal with a large number of gas molecules, we use averages for both speed and kinetic energy

Molecular Velocities and Kinetic Energy Example: Calculation of urms Calculate the root-mean-square velocity for a nitrogen molecule at 30 °C

Solution: Convert the temperature into Kelvin: 30°C+273=303 K Determine the molar mass of nitrogen in kilograms: 28.0g / 1 mol × 1 kg / 1000g = 0.028kg/mol Replace the variables and constants in the root-mean-square velocity equation, replacing Joules with the equivalent kg m^2s^-2: 𝑢rms = √3𝑅𝑇/𝑀 𝑢rms = √ (3(8.314J/mol K)(303 K)) / (0.028kg/mol) = √2.70×10^5m^2s^−2 = 519m/s

The various gas laws can be derived from the assumptions of the KMT, which have led chemists to believe that the assumptions of the theory

accurately represent the properties of gas molecules

The test of the KMT and its postulates is its ability to

explain and describe the behavior of a gas

Molecular Velocities and Kinetic Energy If the temperature of a gas increases, its KEavg increases, more molecules have higher speeds and fewer molecules have lower speeds, and the distribution shifts toward

higher speeds overall, that is, to the right

Kinetic Molecular Theory (KMT)

is a simple microscopic model that effectively explains the gas laws (A model used to explain the behavior of gases in terms of the motion of their particles)

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I: The molecular speed distribution for nitrogen gas (N2) shifts to the right and flattens as the temperature increases;

it shifts to the left and heightens as the temperature decreases

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I: Molecular velocity is directly related to

molecular mass. At a given temperature, lighter molecules move faster on average than heavier molecules

At a given temperature, all gases have the same KEavg for their molecules. Gases composed of lighter molecules have

more high-speed particles and a higher urms, with a speed distribution that peaks at relatively higher velocities

Gases consisting of heavier molecules have

more low-speed particles, a lower urms, and a speed distribution that peaks at relatively lower velocities

The Kinetic-Molecular Theory Explains the Behavior of Gases, Part II: The rate of effusion of a gas depends directly on the (average) speed of its molecules: effusion rate ∝ 𝑢rms Using this relation, and the equation relating molecular speed to mass, Graham's law may be easily derived as shown here:

𝑢rms = √3𝑅𝑇 /𝑀 𝑀= 3𝑅𝑇 / 𝑢^2 rms = 3𝑅𝑇/𝑢^2 effusion rate A / effusion rate B = 𝑢rmsA / 𝑢rmsB = (√3𝑅𝑇/𝑀A ) / (√3𝑅𝑇/𝑀B) = √𝑀B/𝑀A The ratio of the rates of effusion is thus derived to be inversely proportional to the ratio of the square roots of their masses. This is the same relation observed experimentally and expressed as Graham's law

In the KMT, the root mean square velocity of a particle, urms, is defined as the square root of the average of the squares of the velocities with n = the number of particles:

𝑢rms = √𝑢2 = √𝑢^2 1+𝑢^2 2+𝑢^2 3+𝑢^2 4+.../ n *View Picture!