5.1.4 Ideal gases

The mole

The number of atoms/molecules in a given volume of gas can be expressed in terms of moles. This indicates the number of elementary entities within the sample. One mole is the amount of a substance that contains as many elementary entities as there are atoms in 12g of carbon-12. This number is the Avogadro constant (6.02214086 × 10²³ mol^-1). The total number of atoms/molecules in a substance, N, can be calculated using the equation...

N = n x Nₐ

The molar mass of a substance is the mass of one mole of that substance (pretty self explanatory). It can be calculated using the equation...

m = n x M

Kinetic theory of gases

The kinetic theory of gases is a model used to describe the behavior of the atoms or molecules in an ideal gas. There are a number of assumptions made about the atoms/molecules in an ideal gas...

• the gas contains a large number of atoms moving in a random motion with random speeds
• the particles occupy negligible volume compared with the volume of a gas
• all collisions are perfectly elastic (with each other/walls of a container)
• there are negligible electrostatic forces between atoms/molecules except during collisions
• the time of collisions between the atoms/molecules is negligible compared to the time between collisions - okay so this one isn't in the spec but it is in the textbook, so if you only learn 4 learn the above 4, otherwise learn all 5:)

Newton's laws of motion combined with these laws allow us to explain how an atom/molecule in an ideal gas causes pressure...

1. Atoms/molecules are always moving in a gas, when they collide with the walls of a container the container exerts a force on them and their momentum changes as they bounce off the wall
2. When a single atom collides with the container wall elastically it's speed doesn't change. It's total change in momentum is -2mu  (2 lots of momentum, because it's velocity changes as it changes direction)
3. According to n2, the force on the atom making frequent collisions between container walls is F(atom) = Δp/Δt. We know that Δp = -2mu (provided the collision is elastic, which is must be as this is one of the assumptions).
4. From N3, the atom exerts an equal but opposite force on the wall
5. A large number of atoms are colliding randomly with the walls of the container. The pressure they exert on the wall is P = F(total) / A where A is the cross-sectional area of the wall.

Gas laws

Pressure and volume
Provided the temperature and mass of a gas remain constant, the pressure of an ideal gas is inversely proportional to its volume. Therefore, pV = constant. This is Boyle's law.

Pressure and temperature
Provided the volume and mass of a gas remain constant, the pressure of an ideal gas is directly proportional to it's absolute temperature (this is the temperature in Kelvin, not celcius). therefore, p/T = constant.

We can estimate absolute zero using the above equation...

• Connect a flask of dry air to a pressure gauge and introduce it to a water bath
• Increase the temperature of the water bath (measure this using a thermometer)
• Record the resulting increase in pressure of the gas inside the sealed flask using the pressure gauge
• Plot a graph of pressure against temperature (in °C)
• Extrapolate the like back to where p = 0. At absolute zero the particles are not moving (the internal energy is at it's minimum so the pressure of the gas must be zero). This means at p = 0, the temperature is absolute temperature.

Combined laws

We can combine the two above constant equations to give pV/T = constant (for one mole of an idal gas this constant is called the molar gas constant, R, 8.3144598 m² kg s^-2 K^-1 mol^-1). This means that before and after a change of conditions...

p1V1/T1 = p2V2/T2

This leads me on to the equation of state of an ideal gas. For n moles of gas the equation can be rewritten as...

pV = nRT

If we were to plot pV against T, the gradient of this graph would be nR. Considering R is constant. The greater the number of moles means the steeper the gradient.

Root mean square speed

In order to describe the typical motion of particles inside a gas we use the r.m.s. speed. This is because all the velocities of such a large number of particles would cancel each other out. In order to determine r.m.s. speed we square the velocity of each atom/molecule in the gas then take the average of this for all the gas molecules then we take the square root of this.

We need to be able to determine the r.m.s. speed as we use it in the equation...

pV = (NmC²x̅)/3

NOTE: C²x̅ is my attempt at mean square speed on a computer (it is NOT root mean square speed).

The Maxwell-Boltzmann distribution and the Boltzmann constant

The r.m.s. speed is ultimately just an average. At any temperature the random motion of particles means that some are travelling faster than others. The range of speeds of the particles in a gas at a given temperature is known as the Maxwell-Boltzmann distribution...

The hotter the gas the greater the range of speeds and the modal speed and r.m.s. speed increase and the distribution spreads out more.

The Boltzmann constant (k) is used to relate the mean kinetic energy of atoms/molecules in a gas to temperature. It is equal to the molar gas constant (R) divided by the Avogadro constant (Nₐ). In other words, it is 1.38 × 10^-23 m² kg s^-2 K^-1(J K^-1). Using the Boltzmann constant, we can express the equation of state like this instead...

pV = NkT

This is because...

1. You can substitude the definition of the Boltzmann constant (k) into the pV = nRT equation to give pV = nkNₐT
2. The number of particles in a gas, N, = n x Nₐ

Furthermore, we can combine the equations pV = NkT and pV = (NmC²x̅)/3 and form another equation that relates the mean kinetic energy of the particles in a gas to the absolute temperature of that gas...

They both equate to pV so they equate to each other...

(NmC²x̅)/3 = NkT

cancel N...

(NC²x̅)/3 = kT

multiply by 3/2...

(NC²x̅)/2 = 3(kT)/2

tah-dah.

This equation relates Ek (mean average kinetic energy, NC²x̅) to T (the absolute temperature) since all the other values are constant. This means that at a given temperature all the atoms/molecules around us have the same kinetic energy despite the masses of their molecules. This means they must have different r.m.s. speeds.

NOTE: This only works with the absolute temperature which is the temperature in Kelvin.

Internal energy of an ideal gas

The internal energy of a gas is the sum of all the kinetic and potential energies of the particles inside the gas. One assumption (see above) of an ideal gas is that the electrostatic forces between particles in the gas are negligible except during collisions. This means there is no electrical potential energy in an ideal gas so all the internal energy is in the form of the kinetic energy of the particles. This means that doubling the temperature of an ideal gas doubles the average kinetic energy of the particles inside the gas and therefore doubles its internal energy etc (they are proportional).