Gas Laws and Solutions: Gas Laws

Kinetic Molecular Theory (KMT)

The kinetic molecular theory is based on the assumption of an ideal, or “perfect,” gas. Although the perfect gas does not exist in the real world, the kinetic molecular theory is describing this imaginary gas and the KMT is extremely useful for predicting gas behavior. Remember, the KMT is a theory. It has been proven many times, but it is not absolute. It is an attempt to treat all real gases as if they were the same and their differing behaviors are approximated by the ideal gas.

• A gas consists of objects with a defined mass and zero volume.
  This is based on the idea that all real gas particles are of extremely small volume. To apply this to real gases, it is assumed that they are all the same volume and that volume is zero. The masses of real gases are usually small and this allows for them to have a defined mass of zero also.
• The gas particles travel randomly in straight-line motion where their movement can be described by the fundamental laws of mechanics.
  Gas particles are always in motion except at absolute zero (0 K). The particles will travel in random straight line paths. The only deviation from this straight line path is collisions with other particles. If a gas is in a closed system, then it will expand outward to fit the larger volume of its container.
• All collisions involving gas particles are elastic. No energy is lost and no heat is produced.
  An elastic collision is one in which there is no loss of kinetic energy. While the gas phase system undergoes elastic collisions, it is possible for the distribution of the kinetic energy in the system to change. That is, kinetic energy can be transferred; some objects can gain kinetic energy, but only if others lose kinetic energy. The important point is that the total kinetic energy of the system remains constant.
• The gas particles do not interact with each other or with the walls of their container.
  Even though the gas particles collide with each other, they do not react with each other. This also applies to the container’s walls. No bonds or attractions can form, if they did, this would lead to restricted motion of the gas particles.
• Temperature is directly proportional to the kinetic energy of the gas phase system.
  If the temperature of a gas remains constant then the average kinetic energy of the system will also remain constant.

Ideal Gases

The ideal gas model states:

• Ideal gas particles are so small that the volume of the individual particles if they were at rest would be essentially zero when compared with the total volume of the gas.
• Ideal gas particles are in constant, rapid, random motion, moving in straight lines in all directions until they collide with other particles or the sides of their container.
• There are no attractive or repulsive forces between particles, and collisions between them are elastic.
• The average kinetic energy of the particles is directly proportional to the absolute temperature (measured in Kelvin).

Real Gases

Real gases are defined as gases that do not fit the kinetic molecular theory. Even one deviation places the gas in the real gas category. The KMT will always cause problems for scientists, after all, it is a theory and still being tested. However, two main characteristics cause deviations from ideal behavior:

• Very large volume gas molecules will deviate significantly from ideal behavior. Since an ideal gas is able to move anywhere in the container, large volume particles will be restricted from parts of the container that are occupied by the other gas particles. Ideal gas behavior is based on the volume of a gas being very small. Theoretically, helium is the smallest volume gas commonly found in our environment.
• Gas particles that exhibit polar qualities are going to differ from ideal gas behavior. Any gas that is highly polar, such as water vapor, will experience significant attractions for the other particles in the system. This creates problems with the concept of ideal gases not interacting with each other.

When the system is at high pressure or low temperature it again deviates from ideal behavior. These circumstances cause the particles to be close together and improve the chance of interactions.

The four variables used to describe gases are temperature, pressure, moles, and volume. Temperature is the measure of the average kinetic energy of random motion of the particles in a sample of matter. Because many properties of gases depend on the temperature of the studied system, calculations with gases include a specific temperature. Kelvin is used for all gas law calculations. If you are given a value in ºC, it must be converted to Kelvin.

K = °C + 273
T = t + 273

K = Kelvin
T = absolute temperature
t = Celsius temperature

Pressureis the force exerted per unit area. In relation to gases, pressure is a measure of the total force exerted by the moving particles of a gas as they collide with the walls of the container.

Moles is the fundamental SI unit used to measure the amount of a substance. Moles tell you the quantity of the gas. This expresses the number of objects in the system and does not directly indicate their masses.

Volume is the amount of space an object occupies. Gas particles are widely spaced and the volume of a gas is primarily empty space between particles. As a gas contracts, its particles move closer together. As it expands, they move farther apart.

Graham’s Law of Effusion states that a gas will effuse at a rate that is inversely proportional to the square root of its molecular mass, µ.

r₁ = velocity of gas 1
µ₁ = molecular mass of gas 1
r₂ = velocity of gas 2
µ₂ = molecular mass of gas 2

Sample Problem

An O₂ molecule travels at 480 m/s at room temperature. How fast would a molecule of SO₃ travel at the same temperature?

First, list the given values:
r₁ = 48 0 m/s
µ₁ = 32 g/mol (16 g/mol × 2)
µ₂ = 80 g/mol (32 g/mol + 16 g/mol × 3)

Then, write the equation:

;r₂ = 304 m/s

Dalton’s Law states that the sum of the individual pressures of all the gases that make up a mixture is equal to the total pressure.

P_T = P₁ + P₂ + P₃ + …, or

Sample Problem

A 32.0 mL sample of H₂ gas collected over water has a pressure of 750.0 torr. What is the partial pressure of H₂ gas if the total pressure is 875.5 torr?

P₁ = 750.0 torr
P₂ = P 2
P_T = 875.5 torr

P_T = P₁ + P₂ +
P₃ + …..
875.5 torr = 750.0 torr + P₂
P₂ = 125.5 torr

Charles’s Law states that the volume of a gas varies directly with the Kelvin temperature, assuming that the pressure is constant. As one goes up, the other goes up.

Sample Problem

A sample of nitrogen occupies a volume of 250.0 mL at 298 K. What volume will it occupy at 368 K?

V₁ = 250 mL
T₁ = 298 K
T₂= 368 K

V₂ = 309 mL

Boyle’s Law states that the volume of a gas varies inversely with its pressure if temperature is held constant. As one goes up, the other goes down.

P₁ × V₁ = P₂ × V₂

Sample Problem

A sample of CO₂ gas occupies a volume of 3.50 L at 125 kPa. What pressure would the gas exert if the volume were decreased to 2.0 L?

P₁ = 125 kPa
V₁ = 3.50 L
V₂ = 2.0 L

P₁ × V₁ = P₂ × V₂
125 kPa × 3.50 L = P₂ × 2.0 L
P₂ = 220 kPa

Ideal Gas Law uses the four gas variables in relationship to each other. Remember temperature is always calculated in Kelvin.

PV = nRT
R = 0.0821 L∙atm/mol∙K
R = 8.314 L∙kPa/mol∙K

Sample Problem

How many moles of oxygen will occupy a volume of 2.5 L at 1.2 atm and 298 K?

P = 1.2 atm
V = 2.5 L

PV = nRT
1.2 atm × 2.5 L = n × (0.0821 L∙atm/mol∙K) × 298 K
n = 0.12 mol

Be sure to use the same pressure units for the problem and the R constant.

Review of Gas Laws