An ideal gas is a gas where the atoms do not exert forces on each other but they do collide with the walls of the container (in elastic collisions). Based on common sense and experiment the ideal gas law relates the pressure, temperature, volume, and number of moles of ideal gas:

PV = nRT,

where R is a constant known as the universal gas constant.

Comments:

1. Be careful all quantities are expressed in the same system of units!
2. The temperature, T, must be expressed as an absolute temperature, Kelvin.
3. n is the number of moles of the gas, defined as

n º m_sample/M

M º N_A m_atom

where m_sample is the mass of the whole gas sample, N_A is Avogadro’s number and m_atom is the mass of one atom of the gas. M is called the molecular mass of the gas (the mass of one mole of the gas). If by N we mean the total number of atoms of the gas then we also can write,

n = N/N_A

This implies 1 mole consists of N_A atoms of the gas.

In class I will prove based on Newton’s second law and the ideal gas law

E_int = 3/2 n R T (for a monatomic ideal gas = "m.i.g.")

Therefore,

1. the internal energy of an ideal gas depends only on its absolute temperature, and
2. temperature is a measure of the random kinetic energy of atoms.
3. This equation is a remarkable equation since it provides a connection between the macroscopic world (n, T) and the microscopic world (E_int of a gas of atoms)

Since the internal energy of a m.i.g. is entirely kinetic we have also

E_int = ½ m_sample <v²> = 3/2 n R T,

which gives the root mean square speed of an atom of the gas

v_RMS = [ 3RT/M]^1/2 .

Caution: Keep straight n, N, N_A, m_atom, m_sample, and M. These are six distinct quantities.

[in class]