4.2.1. Ideal Gas Law

Table of Contents

From Gas Behavior to a Simple Law

The ideal gas law is a compact equation that connects the main measurable properties of a gas. It tells us how pressure, volume, temperature, and amount of gas are related. For many ordinary gases, especially at low pressure and moderate temperature, this law gives a very good approximation.

The equation is

$$PV = nRT$$

Here, $P$ is the pressure of the gas, $V$ is its volume, $n$ is the amount of substance in moles, $R$ is the universal gas constant, and $T$ is the absolute temperature.

This law is called the ideal gas law because it describes an idealized gas, a gas whose particles are treated in a very simple way. The deeper microscopic picture belongs to other topics, so here we focus on how to use the law and what it means at the macroscopic level.

Important ideal gas law:
$$PV = nRT$$
This equation only works when temperature is measured in kelvin, not in degrees Celsius.

Meaning of Each Quantity

Pressure $P$ measures how strongly the gas pushes on the walls of its container. Volume $V$ is the space occupied by the gas. Temperature $T$ tells us the thermal state of the gas, and in this law it must be the absolute temperature in kelvin. The quantity $n$ measures how much gas is present, in moles.

The gas constant $R$ has the value

$$R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$$

You may also see other numerical forms of $R$ when different pressure and volume units are used, such as liters and atmospheres.

Quantity	Meaning	Common SI Unit
$P$	Pressure	Pa
$V$	Volume	m$^3$
$n$	Amount of gas	mol
$R$	Gas constant	J mol$^{-1}$ K$^{-1}$
$T$	Absolute temperature	K

Why Kelvin Must Be Used

The ideal gas law uses absolute temperature. Absolute zero is the temperature at which thermal motion reaches its minimum possible limit. On the Celsius scale, this is $-273.15^\circ\text{C}$. Because of this, temperature in the formula must be written as

$$T(\text{K}) = T(^\circ\text{C}) + 273.15$$

If Celsius is used directly, the equation gives meaningless results.

For example, $20^\circ\text{C}$ is not $20\ \text{K}$. It is

$$20 + 273.15 = 293.15\ \text{K}$$

Always convert temperature to kelvin before using
$$PV = nRT$$

Rearranging the Law

The ideal gas law can be solved for any one of the four variables if the others are known.

For pressure,

$$P = \frac{nRT}{V}$$

For volume,

$$V = \frac{nRT}{P}$$

For amount of gas,

$$n = \frac{PV}{RT}$$

For temperature,

$$T = \frac{PV}{nR}$$

These rearrangements are useful in practical problems. If a gas is compressed into a smaller volume while the amount and temperature stay fixed, the pressure must increase. If the temperature rises while the amount and volume stay fixed, the pressure must also rise.

Visualizing the Relationship

The ideal gas law shows that gas behavior depends on the balance among four quantities. If one quantity changes while the others are fixed, another quantity must adjust.

Rendering scientific sketch…

Pressure, volume, temperature, and amount in an ideal gas

Units and Consistency

In SI units, pressure is measured in pascals and volume in cubic meters. Then $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$ works directly.

Sometimes gas problems use liters and atmospheres. In that case, a different numerical value of $R$ is often used:

$$R \approx 0.0821\ \text{L atm mol}^{-1}\text{K}^{-1}$$

The key rule is to keep units consistent.

Pressure Unit	Volume Unit	Useful Form of $R$
Pa	m$^3$	$8.314\ \text{J mol}^{-1}\text{K}^{-1}$
atm	L	$0.0821\ \text{L atm mol}^{-1}\text{K}^{-1}$

Since $1\ \text{J} = 1\ \text{Pa}\cdot \text{m}^3$, the SI form is fully consistent.

Use one consistent set of units throughout a calculation. Mixing units carelessly is one of the most common mistakes in gas law problems.

Amount of Gas and Avogadro's Idea

The symbol $n$ gives the amount of gas in moles. A mole is a counting unit for a very large number of particles. If the number of particles is $N$, then

$$n = \frac{N}{N_A}$$

where $N_A$ is Avogadro's constant,

$$N_A \approx 6.022 \times 10^{23}\ \text{mol}^{-1}$$

This means the ideal gas law can also be written in terms of the number of particles:

$$PV = Nk_B T$$

where $k_B$ is Boltzmann's constant. This form connects the macroscopic gas law to the particle description of matter.

Special Cases

When the amount of gas stays fixed, the ideal gas law becomes a relationship among $P$, $V$, and $T$ only:

$$\frac{PV}{T} = \text{constant}$$

This is very useful because it combines several simpler gas laws into one expression. For a fixed amount of gas,

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$

This equation lets us compare an initial state and a final state of the same gas sample.

For example, if a gas is heated while kept in the same volume, then pressure rises in direct proportion to temperature. If a gas expands at constant temperature, then pressure falls as volume increases.

Worked Example

Suppose $2.0$ moles of an ideal gas are in a container of volume $0.050\ \text{m}^3$ at temperature $300\ \text{K}$. What is the pressure?

Using

$$P = \frac{nRT}{V}$$

we substitute the values:

$$P = \frac{(2.0)(8.314)(300)}{0.050}$$

$$P = 9.98 \times 10^4\ \text{Pa}$$

So the pressure is approximately

$$P \approx 1.0 \times 10^5\ \text{Pa}$$

This is about one atmosphere.

Another Example with State Change

A gas initially has pressure $P_1 = 1.0\ \text{atm}$, volume $V_1 = 2.0\ \text{L}$, and temperature $T_1 = 300\ \text{K}$. It changes to a new volume $V_2 = 3.0\ \text{L}$ and temperature $T_2 = 450\ \text{K}$. Find the new pressure.

Use

$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$

Solve for $P_2$:

$$P_2 = P_1 \frac{V_1}{V_2}\frac{T_2}{T_1}$$

Substitute values:

$$P_2 = (1.0)\frac{2.0}{3.0}\frac{450}{300}$$

$$P_2 = 1.0\ \text{atm}$$

So in this case, the increased temperature exactly balances the larger volume, and the pressure remains the same.

What the Law Predicts Qualitatively

The ideal gas law gives several simple predictions.

If $n$ and $T$ are constant, then

$$P \propto \frac{1}{V}$$

If $n$ and $V$ are constant, then

$$P \propto T$$

If $P$ and $T$ are constant, then

$$V \propto n$$

If $P$ and $V$ are constant, then

$$n \propto \frac{1}{T}$$

These proportionalities help you understand gas behavior before doing a full calculation.

Limits of the Ideal Gas Law

Real gases do not always behave ideally. The ideal gas law is most accurate when gas particles are far apart and interactions between them are small. At very high pressures or very low temperatures, real gases can deviate significantly from this simple equation.

Even so, the ideal gas law is one of the most important equations in thermodynamics because it works well in many everyday situations and gives a clear first model of gas behavior.

The ideal gas law is an approximation for real gases. It works best for dilute gases, moderate temperatures, and pressures that are not too high.

Summary

The ideal gas law relates pressure, volume, temperature, and amount of gas through