13.3 The Ideal Gas Law

Ideal Gas Law

The ideal gas law connects pressure, volume, temperature, and the amount of gas in a single equation. It lets you predict how a gas responds when any of those variables change, and it's the foundation for several named gas laws you'll encounter in this unit.

Ideal Gas Law Representations

There are two ways to write the ideal gas law, depending on whether you're counting individual molecules or moles.

Molecular form:

$PV = NkT$

• $P$ = pressure of the gas (in pascals, Pa)
• $V$ = volume of the gas (in cubic meters, m³)
• $N$ = total number of gas molecules
• $k$ = Boltzmann constant = $1.38 \times 10^{-23}$ J/K
• $T$ = absolute temperature (in Kelvin, K)

Molar form:

$PV = nRT$

• $n$ = number of moles of gas
• $R$ = universal gas constant = $8.314$ J/(mol·K)
• $P$, $V$, and $T$ mean the same thing as above

Both forms describe the same physics. You pick whichever matches the information you're given. If a problem tells you the number of molecules, use the molecular form. If it gives you moles, use the molar form.

Applications of the Ideal Gas Law

The ideal gas law contains several simpler laws as special cases. Each one holds a different set of variables constant.

• Boyle's law (constant $T$ and $n$): $P_1V_1 = P_2V_2$. If you double the volume, pressure drops to half.
• Charles's law (constant $P$ and $n$): $\frac{V_1}{T_1} = \frac{V_2}{T_2}$. If you double the Kelvin temperature, the volume doubles.
• Gay-Lussac's law (constant $V$ and $n$): $\frac{P_1}{T_1} = \frac{P_2}{T_2}$. Heating a rigid container raises the pressure.
• Combined gas law (constant $n$, but $P$, $V$, and $T$ all change): $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

The subscripts 1 and 2 refer to the initial and final states of the gas.

How to use these in a problem:

1. Identify which variables stay constant and which change.
2. Pick the law that matches (or use the full ideal gas law if nothing simplifies).
3. Plug in your known values, making sure temperature is in Kelvin.
4. Solve for the unknown.

When multiple variables change at once, you can reason directly from $PV = nRT$. For example, if both pressure and volume double (and $n$ stays fixed), then the left side $PV$ has quadrupled, so $T$ must also quadruple. Or if pressure triples and volume is cut in half, the product $PV$ is multiplied by $3 \times \frac{1}{2} = \frac{3}{2}$, so temperature must increase by a factor of $\frac{3}{2}$.

Dalton's law of partial pressures also follows from the ideal gas law: in a mixture of gases, the total pressure equals the sum of the pressures each gas would exert on its own. Each gas contributes a partial pressure proportional to its number of moles.

Molecule–Mole Conversions in Gas Laws

Avogadro's number ($N_A = 6.022 \times 10^{23}$ molecules/mol) bridges the two forms of the ideal gas law.

• Molecules to moles: $n = \frac{N}{N_A}$
• Moles to molecules: $N = n \times N_A$

Example 1: You have $2.5 \times 10^{24}$ molecules of helium. How many moles is that?

$n = \frac{2.5 \times 10^{24}}{6.022 \times 10^{23}} = 4.15 \text{ mol}$

Example 2: You have 2.7 moles of nitrogen. How many molecules?

$N = 2.7 \times 6.022 \times 10^{23} = 1.63 \times 10^{24} \text{ molecules}$

If a problem gives you molecules but you want to use $PV = nRT$, just convert first: substitute $n = \frac{N}{N_A}$ to get $PV = \frac{N}{N_A}RT$. The same idea works in reverse if you need to go from moles to the molecular form.

Theoretical Foundations and Limitations

The ideal gas law assumes gas molecules are tiny point particles that don't attract or repel each other and don't take up any space themselves. The kinetic theory of gases provides the microscopic justification for this: it models gas molecules as randomly moving particles whose collisions with container walls produce pressure.

Real gases follow the ideal gas law closely at high temperatures and low pressures, where molecules are far apart and moving fast. They deviate when:

• Pressure is very high (molecules are squeezed close together, so their actual volume matters)
• Temperature is very low (molecules slow down enough for attractive forces between them to matter)

The van der Waals equation corrects for both of these effects, but for most problems in an introductory course, the ideal gas law works well.