Finding Mr from gas data

4 minAS Level4.1.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find the relative molecular mass, $M_r$ , of a gas from its mass, pressure, volume and temperature.

The idea: weigh the gas, count the moles

Moles are the bridge between mass and formula:

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

where $m$ is the mass in grams and $M$ is the molar mass in $\text{g mol}^{-1}$ .

The relative molecular mass, $M_r$ , is the same number as $M$ without units — the mass of one molecule relative to $\dfrac{1}{12}$ the mass of a carbon-12 atom.

So if we can measure a gas's mass and its moles, we can find $M_r$ .

Why gas data gives moles so easily

Counting moles of a solid means weighing it and knowing $M_r$ — which is what we want! With a gas there is a shortcut: $pV = nRT$ counts the moles for you from $p$ , $V$ and $T$ .

Then $M_r = \dfrac{m}{n}$ : the mass you weighed, divided by the moles the gas equation counted.

That is the whole trick: measure $m$ , $p$ , $V$ , $T$ ; get $n$ from $pV = nRT$ ; then $M_r = \dfrac{m}{n}$ .

Definition

Working formula for Mr

Combining $n = \dfrac{m}{M_r}$ with $pV = nRT$ gives:

$M_r = \dfrac{mRT}{pV}$

with $m$ in grams, $p$ and $V$ in your chosen recipe's units ( $\text{kPa}$ and $\text{dm}^3$ , or $\text{Pa}$ and $\text{m}^3$ ), and $T$ in kelvin.

Worked example: Mr of a gas from a syringe

A gas syringe holds $120\ \text{cm}^3$ of a gas at $100\ \text{kPa}$ and $25\ ^\circ\text{C}$ . The gas in the syringe has a mass of $0.140\ \text{g}$ . Calculate $M_r$ for the gas. ( $R = 8.31\ \text{J K}^{-1}\text{mol}^{-1}$ )

Have a go first?

Worked example: Mr from gas density

At $100\ \text{kPa}$ and $25\ ^\circ\text{C}$ , a gas has a density of $1.78\ \text{g dm}^{-3}$ . Calculate its $M_r$ .

Have a go first?

Common mistakes

Use the mass of the container too.

Only the gas's mass goes in: usually (mass of flask + gas) − (mass of empty flask).

Put the mass in kg into $M_r = \dfrac{mRT}{pV}$ with $\text{kPa}$ and $\text{dm}^3$ .

Try it yourself

CalculateQuestion 1 · Apply

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