Gas Laws

A2 Physics

Gas Laws

As gas is heated it becomes less dense, i.e. its volume increases, causing the balloon to rise.

Contents:

    Brownian motion
    Pressure
    Boyle's law (constant temperature)
    Charles's law (constant pressure)
    Pressure law (constant volume)
    The equations
    The ideal gas equation
    Avagadro's number
    Kinetic theory and molecular speeds
    Ideal gas assumptions

Brownian Motion

This is the random motion of particles. It can be modeled mathematically and can tell us a lot about the way in which particles behave, particularly gases and liquids.

A smoke cell can demonstrate Brownian motion very well. Smoke particles are very light and their direction of motion can be affected by air molecules..When there is a net momentum in one direction the smoke particles will move in that direction. Because the motion of particles in a gas is random the smoke particles will appear to be moving in random directions.

This is why Brownian motion is only observed on a large enough scale.

Simulation of gas molecules undergoing Brownian motion. (Wikipedia)

Simulation of a dust/smoke particle moving through a gas. (Wikipedia)

Pressure, P

Pressure, P, of a gas is the force, F per unit area, A. It is measured in Pascals (Pa) where 1Pa = 1Nm^-2.

Boyle's Law (constant temperature)

Boyle's law explains how the pressure and volume of a gas vary when the temperature remains constant.. A change at constant temperature is called an isothermal change. Boyle's law states:

The graph for this inverse relationship is as follows:

Thanks Wikipedia

The pressure of a gas at constant temperature is increased by reducing its volume because gas molecules travel less distance between impacts at the walls due to the reduced volume. This means there are more impacts per second, so the pressure is greater.

Charles's Law (constant pressure)

Charles's law explains how the temperature and volume of a gas vary when the pressure remains constant. Charles's law states:

The graph for this relationship is as follows:

Thanks Wikipedia.

As the temperature increases the volume must increase, this is because (for constant pressure) if the particles are gaining kinetic energy with temperature, for them to have the same number of wall collisions per second (i.e. pressure) the walls of the container need to get further apart i.e. it expands.

Pressure Law (constant volume)

The pressure law explains how the pressure and temperature of a gas vary when the volume remains constant. The pressure law states:

The graph for this relationship is as follows:

Once the volume is fixed if you increase the temperature you are giving more energy to the particles in the gas. By doing so their kinetic energies increase which will make them hit the walls of their container more frequently, causing the pressure to rise as well.

The Equations

The above 3 laws reduce to the following:

For the purpose of calculations the following form is convenient where 'i' represents initial conditions and 'f' represents final conditions:

In summary:

Boyle's law = constant temperature

Charles' law = constant pressure

Pressure law = constant volume

The Ideal Gas Equation

How P, V and T are related:

P = pressure (Pa)
V = volume (m^3)
T = temperature (K)
n = number of moles
N = number of molecules
R = universal gas constant = 8.3145 Jmol^-1K^-1
kB = Boltzmann's constant = 1.38x10-23 JK^-1
NA = Avagadro's number = 6.023x1023 mol^-1

Avagadro's Number

Avagadro's law states that equal volumes of different gases at the same pressure and temperature will contain equal numbers of particles. This number is NA = 6.023x10^23 mol^-1

Kinetic Theory and Molecular Speeds

Molecules in an ideal gas have a continuous spread of speeds. The speed of a molecule can change when it collides with another. However, the distribution remains the same provided the temperature is constant.

Ideal Gas Assumptions

There are various versions of the gas laws and some are the same version of the others but said in a slightly different way:

1. All molecules of a particular gas are identical.
2. The internal energy of the gas is entirely kinetic.
3. All collisions between molecules and the walls of the container are completely elastic.
4. Newton’s laws of motion apply.
5. Molecules take up negligible volume.
6. Gravitational and electrostatic forces can be ignored.
7. The motion of all molecules is random.
8. All molecules travel in straight lines.

Molecules are travelling in all directions (+ve and -ve). For the most meaningful mean of the speed of the molecules we must take the root mean square, rms. The rms speed of molecules in an ideal gas, this gives a mean of the magnitude of the speeds:

The rms speed of molecules can be used to link pressure and speed of molecules. This is sometimes referred to as the kinetic theory equation:

The mean kinetic energy of one molecule in a gas is given by the equation (NB this is independent of the mass of each molecule):

The total energy for n moles of an ideal gas is: