The Ideal Gas Equation For Molecules

If you are just looking for the equation, it is: $pV = NkT$.

Quick Recap:

  • The equation for one mole of an ideal gas is $pV = nRT$
  • $N = nN_{A}$ where $N$ = number of molecules, $n$ = number of moles and $N_{A}$ is Avogadro’s constant.

…Applying This:

Rearranging $N = nN_{A}$ for $n$ gives us:

$$\large{n = \frac{N}{N_{A}}}$$

If we substitute this fraction $\frac{N}{N_{A}}$ into $pV = nRT$ we get:

$$\LARGE{pV = \frac{N}{N_A} RT}$$

Switching $N$ and $R$ (we can do this since they are all being multiplied by eachother) we get the form $pV = N\frac{R}{N_A}T$.

We now uncover a new constant:


This is known as the Boltzmann Constant, denoted $k$ and with value $1.38 \times 10^{-23}\space JK^{-1}$

Now we can replace the $\frac{R}{N_A}$ with $k$ to give:

$$\LARGE{pV = NkT}$$


$p$ - pressure of / exerted by the gas

$V$ - volume occupied by the gas

$N$ - number of molecules of gas in sample

$k$ - Boltzmann Constant ($1.38 \times 10^{-23}\space JK^{-1}$)

$T$ - Absolute temperature of the gas

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