The Ideal Gas Equation For Molecules

If you are just looking for the equation, it is: $p V = N k T$ .

The equation for one mole of an ideal gas is $p V = n R T$
$N = n N_{A}$ where $N$ = number of molecules, $n$ = number of moles and $N_{A}$ is Avogadro’s constant.

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

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

If we substitute this fraction $\frac{N}{N_{A}}$ into $p V = n R T$ we get:

$p V = \frac{N}{N_{A}} R T$

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

We now uncover a new constant:

$\frac{R}{N_{A}}$

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

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

$p V = N k T$

Where,

$p$ - pressure of / exerted by the gas

$V$ - volume occupied by the gas

$N$ - number of molecules of gas in sample

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

$T$ - Absolute temperature of the gas

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