Weigh a Liquid, Measure a Gas — and Discover Its Molar Mass

One of the things I particularly enjoy about practical chemistry is that quite sophisticated ideas can sometimes be investigated with surprisingly straightforward apparatus.

A balance.

A syringe.

A gas syringe.

A hot-water bath.

A thermometer.

A pressure reading.

Put them together carefully and we can determine something that sounds as though it ought to require far more elaborate equipment: the molar mass of a volatile liquid.

In this experiment I use methanol. A very small measured mass of liquid methanol is introduced into a heated gas syringe. The methanol evaporates, the gas expands and we measure the volume it occupies at a known temperature and pressure.

From those few measurements, the ideal gas equation allows us to calculate its molar mass.

And, with a little practice and careful technique, the answer can be surprisingly close to the accepted value.

First, a Small but Important Correction: Atomic Mass or Molar Mass?

It is tempting to describe this experiment as measuring the "atomic mass" of methanol.

Strictly speaking, however, methanol is not an atom.

Its formula is:

CH3OH

It is a molecule containing carbon, hydrogen and oxygen atoms.

What this experiment actually determines is its molar mass, expressed in g mol^-1.

We could also compare our result with its relative molecular mass, Mr.

Using the approximate relative atomic masses:

C = 12.01
H = 1.008
O = 16.00

Methanol should therefore have a molar mass of approximately:

12.01 + (4 x 1.008) + 16.00 = 32.04 g mol^-1

The challenge is to see whether we can discover something close to that value experimentally without simply using the periodic table.

The Principle Behind the Experiment

A liquid such as methanol is volatile.

That means its molecules can readily escape from the liquid phase and enter the gas phase.

Methanol has a normal boiling point of about 337.65 K, or approximately 64.5 degrees C, so a water bath maintained comfortably above this temperature can convert a small sample into vapour.

Once the methanol has completely evaporated we can measure four important quantities:

• the mass of methanol, m;

• the volume of methanol vapour, V;

• the gas temperature, T;

• the gas pressure, P.

That gives us everything required for the ideal gas equation:

PV = nRT

where:

P = pressure
V = volume
n = amount of gas in moles
R = gas constant
T = absolute temperature in kelvin

Rearranging:

n = PV / RT

But molar mass is:

M = mass / moles

Therefore:

M = m / n

Substituting our gas equation expression for n gives:

M = mRT / PV

That is the equation at the heart of the entire experiment.

The Apparatus

A typical arrangement might use:

• a gas syringe;

• a thermostatically controlled hot-water bath;

• thermometer or temperature probe;

• a small liquid syringe;

• a good laboratory balance;

• atmospheric pressure measurement;

• clamp and stand;

• suitable injection fitting or septum;

• methanol;

• appropriate personal protective equipment.

The gas syringe should be positioned so that its barrel reaches the temperature of the water bath while its plunger remains free to move.

Ideally, the syringe should also be arranged to minimise the effect of the weight of the plunger on the gas pressure.

This immediately introduces an interesting point that students sometimes overlook:

the apparatus itself can affect the result.

Step One — Establish the Mass of Methanol

A small quantity of methanol is drawn into a suitable syringe.

The syringe containing the methanol is weighed.

Suppose its mass is:

12.486 g

After injecting the methanol, the syringe is weighed again.

Suppose it now reads:

12.406 g

The mass delivered is therefore:

12.486 - 12.406 = 0.080 g

This "weigh by difference" technique is often much better than attempting to measure the mass of the liquid directly.

It also provides a useful lesson in experimental technique.

We are not really interested in the mass of the syringe.

We are interested in the change in its mass.

Step Two — Heat the Gas Syringe

The gas syringe is allowed to reach the temperature of the hot-water bath.

For example, we might use:

75 degrees C

The corresponding absolute temperature is:

75 + 273.15 = 348.15 K

This conversion is essential.

The ideal gas equation must use absolute temperature.

Using 75 rather than 348.15 would produce complete nonsense.

This is a good example of something students can know perfectly well theoretically but still forget when actually performing a calculation.

Step Three — Inject the Methanol

The small measured sample of methanol is injected into the heated gas syringe.

Almost immediately the liquid begins to evaporate.

As molecules enter the gas phase, the volume increases and the syringe plunger moves outwards.

What began as an almost insignificant drop of liquid can occupy tens of cubic centimetres as a gas.

That transformation itself makes quite a good demonstration.

Students often understand intellectually that gases occupy considerably more volume than liquids, but watching a tiny quantity of methanol push a gas-syringe plunger across the scale makes the idea much more tangible.

Step Four — Let Everything Reach Equilibrium

Do not immediately take the first volume reading.

The methanol needs time to evaporate completely.

The vapour needs to reach the temperature of the water bath.

The syringe plunger also needs to settle.

We want the final measurement to represent something close to thermal and mechanical equilibrium.

Suppose the final gas volume is:

71.3 cm3

This is:

0.0713 litres

If the atmospheric pressure is:

101.3 kPa

we now have everything we require.

The Calculation

Our measurements are:

Mass, m = 0.080 g

Temperature, T = 348.15 K

Pressure, P = 101.3 kPa

Volume, V = 0.0713 L

Using:

M = mRT / PV

and:

R = 8.314 kPa L mol^-1 K^-1

we obtain:

M = (0.080 x 8.314 x 348.15) / (101.3 x 0.0713)

M = approximately 32.1 g mol^-1

The accepted value calculated from the molecular formula is approximately:

32.04 g mol^-1

That is an extremely satisfying result.

We have essentially weighed a tiny quantity of liquid, turned it into a gas and used its pressure, volume and temperature to work backwards to the mass of one mole.

Why I Like This Experiment

What I particularly like about this experiment is the way several apparently separate parts of chemistry suddenly become connected.

Students encounter:

• states of matter;

• vaporisation;

• boiling point;

• measurement uncertainty;

• pressure;

• absolute temperature;

• moles;

• molar mass;

• the ideal gas equation.

On paper, these can look like independent topics.

In the laboratory they become one experiment.

The practical also turns PV = nRT from something that can look like an abstract rearrangement exercise into a genuine measuring instrument.

We are using an equation to find something we cannot directly measure.

That, to me, is when equations become most interesting.

Why the Result Is Not Always Perfect

Of course, real experiments rarely behave quite as neatly as worked examples.

The first attempt may produce:

29 g mol^-1.

Or 35 g mol^-1.

Perhaps something considerably worse.

That is not necessarily a failed experiment.

It is an opportunity to investigate why.

Trapped Air

One of the biggest problems is residual air in the gas syringe.

If the syringe initially contains air and we then add methanol vapour, our measured volume does not represent methanol alone.

The calculated number of moles will therefore be wrong.

Careful preparation of the syringe is extremely important.

Incomplete Evaporation

If some liquid remains unevaporated, the measured gas volume represents only part of the sample.

Yet the calculation may assume that the entire measured mass became gas.

This can make the calculated molar mass too high.

The water bath therefore needs to remain sufficiently above the liquid's boiling point and enough time must be allowed for complete evaporation.

Temperature Errors

The thermometer tells us the temperature of the water bath.

But is the gas inside the syringe actually at exactly the same temperature?

Not necessarily.

If the sample is injected and the reading is taken immediately, the gas may not yet have reached thermal equilibrium.

Waiting for a stable volume greatly improves the experiment.

Pressure Errors

We usually assume that the gas pressure in a freely moving gas syringe is approximately equal to atmospheric pressure.

That assumption is not absolutely perfect.

Plunger friction matters.

The orientation of the syringe can matter.

A vertically arranged plunger may require the internal gas to support some of its weight.

A sticky syringe may require a slight excess pressure before it moves.

Good experimental design therefore tries to make the plunger move as freely as possible.

Measuring Such a Small Mass

Suppose the sample weighs only 0.080 g.

An uncertainty of just 0.002 g represents:

0.002 / 0.080 x 100 = 2.5%

That uncertainty alone could shift our final molar mass by roughly the same percentage.

A balance capable of reliably measuring small mass differences makes a considerable improvement.

It also explains why simply making the sample smaller is not always better.

We want a sample small enough to fit comfortably within the gas syringe after vaporisation, but large enough to weigh accurately.

Reading the Gas Syringe

Suppose our gas volume is around 70 cm3.

An error of 1 cm3 is already more than 1%.

Students should therefore read the scale carefully and avoid parallax.

The plunger should also be allowed to settle before taking the reading.

Repeated measurements are much more convincing than a single apparently perfect answer.

Try It More Than Once

This is where the experiment gets interesting.

Rather than performing one measurement and declaring success, repeat it.

For example:

Trial 1: 33.5 g mol^-1
Trial 2: 32.6 g mol^-1
Trial 3: 32.0 g mol^-1
Trial 4: 31.8 g mol^-1

The technique may visibly improve.

Students begin learning how the apparatus behaves.

They learn how long equilibrium takes.

They notice whether the plunger tends to stick.

They become better at injecting the liquid cleanly.

This is genuine practical science.

Skill matters.

Can We Quantify How Good the Result Is?

Suppose our experimental value is:

32.1 g mol^-1

and our expected value is:

32.04 g mol^-1

The percentage difference is approximately:

Percentage difference = |experimental - accepted| / accepted x 100

Therefore:

Percentage difference = |32.1 - 32.04| / 32.04 x 100

= approximately 0.2%

In a real student experiment I would not necessarily expect every attempt to be that good.

But results within a few percent can certainly demonstrate that the method works remarkably well.

An Excellent A-Level Extension: Which Measurement Matters Most?

There is another investigation hidden inside this experiment.

Students could estimate the percentage uncertainty associated with:

• mass;

• volume;

• temperature;

• pressure.

They could then ask:

Which measurement contributes most to the uncertainty in the final molar mass?

Often the mass measurement and gas-volume measurement dominate.

That leads naturally into experimental design.

If we wanted to improve the experiment, where should we spend our money?

A better thermometer?

A better barometer?

A more precise balance?

A larger gas syringe?

Science is not simply about taking measurements.

It is about knowing which measurements are worth improving.

Could We Identify an Unknown Liquid?

Once students understand the method, the experiment can be turned around.

Instead of being told that the liquid is methanol, suppose they are simply given "volatile liquid A".

They determine:

M = approximately 46 g mol^-1

Could it be ethanol?

Another sample gives approximately:

58 g mol^-1

What possible compounds might fit that result?

Suddenly the practical becomes a chemical identification problem.

Molar mass becomes experimental evidence.

Ideal Gases Are Not Actually Ideal

There is also a deeper question for stronger A-level students.

PV = nRT describes an ideal gas.

Real methanol molecules interact with one another.

So why does the experiment work?

Because under suitable conditions the ideal gas equation provides a good enough approximation.

This is an important scientific idea.

Models do not have to describe reality perfectly to be useful.

They need to describe it accurately enough for the question we are asking.

A Word About Safety

Methanol is not simply "another alcohol".

It is highly flammable and is classed as toxic if swallowed, in contact with skin or inhaled; significant exposure can also cause organ damage.

This is therefore a proper supervised laboratory experiment, using very small quantities and an appropriate risk assessment.

In particular:

• there should be no naked flames or ignition sources;

• heating should be by a controlled water bath;

• suitable eye and skin protection should be used;

• vapour exposure should be minimised;

• good ventilation or suitable extraction should be provided;

• spills and waste should be handled correctly.

The small scale of the experiment helps, but small quantity does not mean no hazard.

Safety is part of the chemistry, not something added afterwards.

From a Drop of Liquid to Molecular Information

Perhaps the most impressive part of this experiment is how little information we actually begin with.

We have a small drop of liquid.

We measure its mass.

We turn it into a gas.

We measure its temperature, pressure and volume.

Then mathematics gives us something we cannot see:

the mass of one mole of its molecules.

For methanol we expect about:

32.04 g mol^-1

And with careful experimental technique, a gas syringe and the ideal gas equation can get remarkably close.

That is why I think practical work like this deserves more attention.

It is not simply demonstrating something that students have already learned.

It shows how scientists actually use measurements and models to discover information about matter.

A tiny quantity of colourless liquid becomes a bridge between the macroscopic world we can measure and the molecular world we cannot see.

And all because:

PV = nRT