What the equation connects

The ideal gas equation is:

Each symbol has a precise meaning:

For this course, the value of $R$ is supplied on the data sheet. Use:

The equation treats the gas as ideal under the conditions in the question. You do not need to derive the equation from kinetic theory here. The assessed skill is using the relationship correctly with SI units.

SI units first

The value $R = 8.314\ \text{J mol}^{-1}\text{ K}^{-1}$ works cleanly when pressure is in Pa, volume is in m^3 and temperature is in K. If you substitute kPa, cm^3 or degrees Celsius directly, the arithmetic may look neat but the answer will be wrong.

Use these conversions often:

For example, $75.0\ \text{cm}^3$ is:

And $31^\circ\text{C}$ is:

Worked example: converting a data set

A gas sample is measured at $105\ \text{kPa}$, $250\ \text{cm}^3$ and $27^\circ\text{C}$.

Convert each value before using $pV = nRT$:

The amount $n$ is already in mol if it is given as a number of moles. If the question gives mass instead, find $n$ from the mole calculation already taught, then use the ideal gas equation.

Calculating amount of gas

Many ideal-gas questions ask for the amount of gas, $n$. Start from:

Divide both sides by $RT$:

That rearrangement matters. It also lets an examiner see that you are using the equation, not a gas-volume shortcut.

Worked example: finding $n$

A sample of gas has volume $48.0\ \text{cm}^3$ at $25^\circ\text{C}$ and $100\ \text{kPa}$. Calculate the amount of gas.

First convert:

Now substitute into the rearranged equation:

To three significant figures:

The answer is small, which is reasonable because $48.0\ \text{cm}^3$ is a small gas volume.

Rearranging for other variables

The equation can be rearranged for any one unknown. Do the algebra before substitution where possible, because it reduces calculator mistakes.

From:

You can make these forms:

Worked example 1: finding volume

A sample contains $0.0200\ \text{mol}$ of gas at $40^\circ\text{C}$ and $105\ \text{kPa}$. Calculate the volume in $\text{dm}^3$.

Convert first:

Use:

The question asks for $\text{dm}^3$, so convert back at the end:

Worked example 2: finding pressure

A sample contains $0.0200\ \text{mol}$ of gas in a volume of $650\ \text{cm}^3$ at $25^\circ\text{C}$. Calculate the pressure in kPa.

Convert:

Use:

The question asks for kPa:

Exam habits and traps

A strong ideal-gas solution is not just a final number. It is a clear chain:

1. Write or imply the correct rearranged equation.
2. Convert pressure, volume and temperature to SI units.
3. Substitute values with $R = 8.314$.
4. Calculate without rounding too early.
5. Round the final answer to a sensible number of significant figures and include the correct unit.

The most common errors are small but costly.

Trap 1: using cm^3 or dm^3 directly

If $V = 75.0\ \text{cm}^3$, do not substitute $75.0$. Use:

Trap 2: using degrees Celsius as $T$

If the temperature is $75^\circ\text{C}$, do not use $T = 75$. Use:

Trap 3: choosing the wrong gas-volume method

At room temperature and pressure, some amount-of-substance questions can use molar gas volume. But if a question gives specific pressure and temperature values for an ideal-gas calculation, use $pV = nRT$ with SI units. The supplied $R$ value is a strong clue.

Trap 4: losing the unit at the end

If you calculate $n$, the unit is mol. If you calculate $V$ using SI units, the first answer is in m^3. If the question asks for dm^3 or cm^3, convert after calculating.