Kinetic Theory of Gases

A. Kinetic Molecular Theory Particles in an ideal gas… have no volume. have elastic collisions. are in constant, random, straight-line motion. don’t attract or repel each other. have an avg. KE directly related to Kelvin temperature.

B. Real Gases Particles in a REAL gas… have their own volume attract each other Gas behavior is most ideal… at low pressures at high temperatures in nonpolar atoms/molecules

C. Characteristics of Gases Gases expand to fill any container. random motion, no attraction Gases are fluids (like liquids). no attraction Gases have very low densities. no volume = lots of empty space

C. Characteristics of Gases Gases can be compressed. no volume = lots of empty space Gases undergo diffusion & effusion. random motion

A. Gas Stoichiometry Moles  Liters of a Gas STP - use 22.4 L/mol Non-STP - use ideal gas law Non-STP Problems Given liters of gas? start with ideal gas law Looking for liters of gas? start with stoichiometry conv.

Ptotal = P1 + P2 + ... Dalton’s Law The total pressure of a mixture of gases equals the sum of the partial pressures of the individual gases. Ptotal = P1 + P2 + ... Patm = PH2 + PH2O

B. Graham’s Law Diffusion Spreading of gas molecules throughout a container until evenly distributed. Effusion Passing of gas molecules through a tiny opening in a container

Diffusion The process in which molecules move from a region of higher concentration to one of lower concentration is called diffusion. The host medium, such as the air or water, is referred to as the solvent, while the diffusing substance, like the perfume molecules, is known as the solute. Relatively speaking, diffusion is a slow process, even in a gas.

Why Diffusion Is Relatively Slow In Example 3 we will seen that a gas molecule has a translational rms speed of hundreds of meters per second at room temperature. At such a speed, a molecule could travel across an ordinary room in just a fraction of a second. Yet, it often takes several seconds, and sometimes minutes, for the fragrance of a perfume to reach the other side of a room. Why does it take so long?

KE = ½mv2 Graham’s Law Speed of diffusion/effusion Kinetic energy is determined by the temperature of the gas. At the same temp & KE, heavier molecules move more slowly. Larger m  smaller v because… KE = ½mv2

Graham’s Law Graham’s Law Rate of diffusion of a gas is inversely related to the square root of its molar mass. Ratio of gas A’s speed to gas B’s speed

Kinetic Theory of Gases The pressure that a gas exerts is caused by the impact of its molecules on the walls of the container.

IDEAL GAS LAW The absolute pressure P of an ideal gas is directly proportional to the Kelvin temperature T and the number of moles n of the gas and is inversely proportional to the volume V of the gas: P = R(nT/V). In other words, where R is the universal gas constant and has the value of 8.31 J/(mol·K).

The constant term R/NA is referred to as Boltzmann’s constant, in honor of the Austrian physicist Ludwig Boltzmann (1844–1906), and is represented by the symbol k:

Kinetic Theory of Gases The pressure that a gas exerts is caused by the impact of its molecules on the walls of the container. It can be shown that the average translational kinetic energy of a molecule of an ideal gas is given by, where k is Boltzmann's constant and T is the Kelvin temperature.

Derivation of, Consider a gas molecule colliding elastically with the right wall of the container and rebounding from it.

The force on the molecule is obtained using Newton’s second law as follows,

The force on one of the molecule, According to Newton's law of action–reaction, the force on the wall is equal in magnitude to this value, but oppositely directed. The force exerted on the wall by one molecule,

If N is the total number of molecules, since these particles move randomly in three dimensions, one-third of them on the average strike the right wall. Therefore, the total force is: Vrms = root-mean-square velocity.

Pressure is force per unit area, so the pressure P acting on a wall of area L2 is

Pressure is force per unit area, so the pressure P acting on a wall of area L2 is Since the volume of the box is V = L3, the equation above can be written as,

PV = NkT

Example 2. Oxygen in the Lungs In the lungs, the respiratory membrane separates tiny sacs of air (absolute pressure = 1.00 × 105 Pa) from the blood in the capillaries. These sacs are called alveoli, and it is from them that oxygen enters the blood. The average radius of the alveoli is 0.125 mm, and the air inside contains 14% oxygen. Assuming that the air behaves as an ideal gas at body temperature (310 K), find the number of oxygen molecules in one of the sacs.                     .

Example 3. The Speed of Molecules in Air Air is primarily a mixture of nitrogen N2 (molecular mass = 28.0 u) and oxygen O2 (molecular mass = 32.0 u). Assume that each behaves as an ideal gas and determine the rms speeds of the nitrogen and oxygen molecules when the temperature of the air is 293 K.

The Internal Energy of a Monatomic Ideal Gas