KMT Gas Laws - Lecture notes 4 PDF

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KMT and the Gas Laws Pressure = force per unit area: P = F x A Force = mass times acceleration; F = m x a Kinetic Energy = 1/2 the mass times the square of the velocity ; KE = (1/2)mv2 The key thing about the kinetic energy equation is that it has two components: mass and velocity. Two moving objects of very different mass can have the same KE if they have different velocities. Thus – a lighter gas like hydrogen can have the same KE at a given temperature as heavier oxygen molecules if the ligher hydrogen molecules are moving much faster than the heavier oxygen molecules. Some typical data for a range of

different gases at 273 K (0C) is below to help make this point. Make sure to convince yourself of the truth of these statements before moving on. H2

He

NH3

N2

Ar

F2

CO2

Cl2

MW(g/mol)

2.02

4

17.03

28.02

29.95

38

44.01

70.9

urms, (m/s)

1836

1305

632

493

477

423

393

310

KE (kJ/mol)

3.40

3.41

3.40

3.41

3.41

3.40

3.40

3.41

Gas

Boyle's Law: The pressure of a confined gas at constant temperature is inversely proportional to the volume of the gas. This means as the pressure on a gas increases, the volume will decrease proportionately. If you double the pressure on a gas (x 2) the volume will decrease to one-half of its original value (x 1/2). Mathematically: Pressure times Volume equals a constant: PV = k or P 1/V ( = is proportional to) KMT Explanation: For a gas to exert pressure, it must try and "accelerate" (or move) whatever it collides with. When a gas particle encounters the container wall, it may transfer some of its energy to the wall. This energy transfer represents the "force" applied to the "area" of the container wall. The volume of a gas sample is simply the volume of the container holding the gas. If the volume of the container is increased, the gas expands to fill the new volume and the area covered by the container wall increases. Any force exerted by the gas striking the walls will be spread over a larger area. Each "unit area" segment receives less force and the pressure we record is decreased.

increase volume

In the larger volume – each piece of container wall will experience fewer collisions. Fewer collisions = less force imparted to walls = lower pressure

= a moving gas particle; the tail indicates relative speed.

The energy of the gas comes from the fact that its particles are in continuous motion. Therefore, each gas particle has some kinetic energy. It is also true that sometimes the gas picks up energy from the container wall. Therefore, the overall energy of the gas (and its temperature) remains constant unless the outside temperature is changed.

Gay-Lussac's Law: The pressure of a confined gas at constant volume is directly proportional to the absolute temperature of the gas. Absolute temperature here means Kelvin units. The Kelvin temperature starts at “absolute zero” where the gas particles would have zero kinetic energy. Celsius units are more commonly used in most areas of chemistry, but gases absolutely require absolute temperatures! The direct proportionality means that as the absolute temperature of a gas increases, the pressure will increase proportionately. If you double the absolute temperature of a gas (x 2) the pressure will increase to twice its original value (x 2). Mathematically: Pressure divided by absolute Temperature equals a constant: P/T = k or P  T

KMT Explanation: The increased average kinetic energy of the gas particles caused by the higher temperature (= faster speeds) increases the average force with which the gas particles hit the container walls. If the Force of the collisions increases while the area is constant, the pressure must increase.

The faster particles at higher temperature strike the walls with greater force

increase absolute T

Charles' Law: The volume of a confined gas at constant pressure is directly proportional to the absolute temperature of the gas. The direct proportionality means that as the absolute temperature of a gas increases, the volume will increase proportionately. If you double the absolute temperature of a gas (x 2) the volume will increase to twice its original value ( x 2). Mathematically: volume divided by temperature equals a constant: V/T = k KMT Explanation: Charles’ Law is a consequence of both Boyle’s Law and Gay-Lussac’s Law. If you have a container that can change size (like a balloon) – the pressure of the gas inside will always equal the pressure of air on the outside. If the pressure inside the balloon is equal to the pressure outside the balloon, the balloon remains the same size The outside of the container is constantly being bombarded by air molecules. As long as the gas inside the container and the air outside the container are the same, the size of the container remains constant. If the temperature is raised, the average speed of the gas particles increases, and the average kinetic energy of the gas increases. Higher kinetic energy means that the gas particles will exert greater force when striking the container walls increasing the pressure of the gas. This pushes outward on the walls of the container. If the volume of the container can change this force can be spread over a larger area and the pressure will not have to increase. This is similar to what happens if you heat a sealed balloon. Once the pressure inside the balloon is strong enough to stretch the rubber of the balloon, the balloon increases in size. .

increase absolute T

The pressure of the confined gas is the same as the outside air

volume expands

The faster particles at higher temperature strike the walls with greater force causing increased pressure on the walls

In the larger volume, the greater force is spread over a proportionately larger area and the final pressure is the same as the outside air.

Note: In the real world, the gas inside the balloon must be at a slightly higher pressure than the air since it must also resist the force of the stretched rubber in the balloon.

Dalton's Law of Partial Pressure: The pressure exerted by a gas mixture is equal to the sum of the pressures exerted by the individual gases: Mathematically, Ptotal = pA + pB + … KMT Explanation: The pressure of a gas depends on its average kinetic energy and all gases at the same temperature (as must happen in a mixture) have the same average kinetic energy. The pressure of a gas also depends on the number of particles contained within the volume. The contribution of any individual gas in the mixture to the total pressure will therefore depend on its relative abundance in the mixture (by particle count, not mass). For example, if a gas mixture at a pressure of 800 torr contains 3 units of gas A and 5 units of gas B, gas A will contribute 3/(3+5) x 800 torr = 300 torr of pressure. Gas B will contribute 5/(3+5) x 800 torr = torr of pressure. This is more easily seen if you work the problem from the opposite direction of two gases in the same sized container that are then mixed into one of the containers. In the mixture – each gas continues to more randomly and strike the container walls with the same frequency and force as before since the size of the container is the same as what they started in:

+ Gas A 300 torr pressure

mix gases

Gas B 500 torr pressure

Gas A + Gas B 300 + 500 = 800 torr pressure

Hey! Different gases have different masses, don't they? True, but the pressure that a gas exerts depends on its average kinetic energy which is set by the temperature of the gas. Gases that have different masses can have the same average kinetic energy because two things contribute to the kinetic energy–the mass and the velocity. The heavier a gas is, the slower it will move.

Avogadro's Law: Any gases that that have the same volume under the same conditions of temperature and pressure, MUST contain the same number of individual gas particles. Mathematically, the volume divided by the number of particles (n) equals a constant: V/n = k KMT Explanation: If all gases behave the same with respect to pressure, volume and temperature, the only way that two gases can have the same volume at the same temperature and pressure is to have the same number of particles.

A Final Note about Temperature, Pressure, and Volume changes: While changing the temperature of a gas WILL change the volume and/or pressure of a gas sample, the reverse is NOT NECESSARILY true. This complication arises because the gases can exchange energy with the container walls during collisions. If the change in volume or pressure occurs more rapidly than the gas can exchange energy with the container walls, the temperature of the gas will change. Whether the change is proportional or not depends on exactly how rapid the change is. Conversely – if you change the pressure or volume relatively slowly and the gas particles can exchange energy with the container walls, no temperature change in the gas will occur. For example, a diesel engine uses rapid compression of diesel fuel vapor to ignite the fuel. The fuel vapor is compressed very rapidly and heats up. If the fuel vapor gets hot enough (the autoignition temperature), the vapor will ignite. If you performed this compression very slowly (many minutes perhaps), the increase in the energy of the fuel as a result of the compression would be transferred to the cooler piston walls. The piston walls would, in turn, transfer this energy to the outside air. No temperature change would be seen for the gas or the piston walls. The amount of energy transferred to the atmosphere would be insignificant to the air and no temperature change would be seen for the air either....