Ideal gases

​​In a nutshell

The mole is an SI unit used to measure the amount of a substance. Ideal gases are gases that follow certain assumptions to simplify their behaviour.

Equations

Constants  

Variable definitions

The mole

The mole is an SI unit used to measure the amount of a substance and has the symbol $$mol$$. It is defined as the amount of a substance that contains the number of atoms contained in $$12\ g$$ of carbon-$$12$$, this number being the Avogadro constant $$N_A$$ with a value of $$6.02\times10^{23}\ mol^{-1}$$.​

In other words, $$1\ mol$$ is equal to the amount of a substance that has $$6.02\times10^{23}$$ atoms. To find the number of particles in a substance you can then use the equation:

​$$N=n\times N_A$$​​

Where $$n$$ is the number of moles with units $$mol$$. 

Another quantity associated with the mole is the molar mass $$M$$, with units $$kg\ mol^{-1}$$. It is defined as the mass per $$1\ mol$$ of substance and can be found with the equation:

​$$M=\dfrac{m}{n}$$​​

Example

Calculate the number of atoms in $$20\ g$$ of helium gas. Helium has a molar mass of $$0.004\ kg\ mol^{-1}$$.

Firstly, write down what you know:

$$m=20\ g=0.02\ kg$$

$$M=0.004\ kg\ mol^{-1}$$​​​

Write down the equation for molar mass :

$$M=\dfrac{m}{n}$$

​​​

Rearrange it for $$n$$:

$$n=\dfrac{m}{M}$$​​

Substitute the numbers and calculate the number of moles in $$20\ g$$ of helium gas:

$$n=\dfrac{0.02}{0.004}=5\ mol$$​​

Now write down the equation for the number of atoms in a substance:

$$N=n\times N_A$$​​

Substitute the values and calculate the number of atoms in the helium gas:

$$N=5\times6.02\times10^{23}=3.01\times10^{24}$$​​

There are $$\underline{3\times10^{24}}$$ atoms in $$20\ g$$ of helium gas, to one significant figure. 

The kinetic theory of gases

The kinetic theory of gases is a model used to describe the behaviour of ideal gases. These are different from real gases as they follow a set of assumptions to simplify their behaviour.

The assumptions followed by ideal gases are:

• The gas particles have no intermolecular forces between each other.
  
• The gas particles occupy a negligible volume compared to the gas itself.
• All collisions are perfectly elastic and have negligible time compared to times between collisions, such that no kinetic energy is lost.
• The gas particles move in random directions with random speeds.
  
• There are a very large number of atoms or molecules in the gas.
  

By using these assumptions and Newton's laws it is possible to explain how the particles in a gas behave and cause pressure.

Pressure and change in momentum 

When particles collide with a container wall, they exert a pressure. The pressure can be calculated with the following:

​$$p = \dfrac{F}{A}$$​​

Using Newton's second law, the force that is exerted onto the atoms from the wall, is the rate of change of momentum. The atoms exert an equal but opposite force on the wall. This force can be calculated using the following equation:

​$$F_{atom} = \dfrac{\Delta p}{\Delta t}$$​​

Note: $$\Delta t$$ Is the time between collisions with the wall. Also, don't get confused with $$p$$ for pressure and change in momentum. They mean different things!

The image below shows how the movement of a particle colliding with a wall over time:

1. Positive direction A Before B After

The particle approaching the wall with a momentum $$p=mu$$ bounced back with the same momentum due to the collision being elastic. This means that the total change in momentum is equal to the following:

​$$\Delta p = -2mu$$​​

Note: The reason that the total change in momentum is $$-2mu$$ is due to the velocity changing from $$+ \ u \ ms^{-1}$$ to $$- \ u \ ms^{-1}$$ after it has collided with the wall of a container. This is because it has changed direction.

Example

Calculate the change in momentum when a molecule with mass $$3.0 \times 10^{-26} \ kg$$ collides elastically at right angles with a wall at $$650 \ ms^{-1}$$. 

Firstly, write down what you know:

​$$m = 3.0 \times 10^{-26} \ kg \\ u = 650 \ ms^{-1}$$​

Write down the equation needed:

$$\Delta p = -2mu$$

Substitute in the values, and calculate your final answer to the lowest number of significant figures given in the question: 

$$\Delta p = -2 \times 3.0 \times 10^{-26} \times 650 = -3.9 \times 10 ^{-23} \ kg \ ms^{-1}$$​​

The change in momentum of the molecule is $$\underline {-3.9 \times 10 ^{-23} \ kg \ ms^{-1}}$$.