Charles' law

Leave a sealed balloon on a sunny windowsill and it swells a little; put the same balloon in the fridge and it shrinks. Nothing was added or removed — only the temperature changed, and the balloon's flexible skin let its volume respond while the pressure inside stayed close to atmospheric. That everyday observation is exactly what Charles' Law describes: at constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature. It is the second of the ICSE Class 9 gas laws, coming right after Boyle's Law, and it is the law that forces the idea of absolute zero and the Kelvin scale into full use — you cannot state or apply Charles' Law correctly with Celsius temperatures. This section expands the Pass-1 treatment with the molecular reasoning behind the law, its graphical signature, a full worked derivation of the working formula, and one detailed worked example, so you can move from "remembering the equation" to "knowing why it must be true."

The core insight

At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute (Kelvin) temperature, so $V_1/T_1 = V_2/T_2$ for any two states of the same gas sample.

The core idea

Why heating expands a gas at constant pressure. A gas is a swarm of molecules in constant, random motion, striking the walls of their container and thereby exerting pressure. Heating the gas raises the average kinetic energy of its molecules, so they move faster and strike the walls harder and more often. If the container is rigid, this shows up as higher pressure — but if the container is free to expand (a balloon, or a gas trapped under a movable piston), the gas instead pushes the boundary outward until the molecules are spread over a larger volume, striking the (now larger) wall area just often enough to keep the pressure unchanged. The volume increase is the gas's way of absorbing the extra molecular energy while pressure stays fixed.

The statement of the law. Charles' Law states that at constant pressure, the volume $V$ of a fixed mass of gas is directly proportional to its absolute temperature $T$ :

V \propto T \quad \text{(at constant } P \text{ and constant mass, } T \text{ in kelvin)}

where: $V$ is the gas volume, $T$ is the absolute temperature in kelvin, and the proportionality holds only while pressure and the amount of gas are both fixed. Direct proportionality means $V/T$ is a constant number for that sample of gas at that pressure, so comparing an initial state $(V_1, T_1)$ with a final state $(V_2, T_2)$ gives the working equation:

\frac{V_1}{T_1} = \frac{V_2}{T_2}

Why the Kelvin scale is non-negotiable. If you record volume against Celsius temperature at constant pressure and plot the points, you still get a straight line — but it does not pass through the origin. Extending that line backward, it crosses the temperature axis at $-273\,^{\circ}\text{C}$ , the temperature at which the extrapolated volume would fall to zero. This extrapolated point is absolute zero, the coldest temperature conceivable, and it is precisely where the Kelvin scale places its zero. Once temperatures are measured in kelvin, the same data forms a straight line through the true origin — a genuine direct proportionality, $V \propto T$ . Plugging Celsius values into $V_1/T_1 = V_2/T_2$ breaks the law immediately, because Celsius zero is not the same as "zero volume," and a negative Celsius value would make no physical sense as a ratio denominator.

Figure

Two line graphs of gas volume against temperature one axis in Celsius crossing the temperature axis at minus 273 degrees and the other axis in Kelvin passing straight through the origin

Derivation Aof the Charles' Law equation

The goal of this derivation is to move from the plain-language statement "at constant pressure, volume is directly proportional to absolute temperature" to the usable two-state equation $V_1/T_1 = V_2/T_2$ , showing precisely why the ratio $V/T$ stays fixed for one sample of gas and why that lets you compare any two states of it.

Step 1. Write the proportionality as an equation. Direct proportionality $V \propto T$ means there exists some constant $k$ , fixed for a given mass of gas at a given pressure, such that $V = kT$ . This is the general algebraic meaning of "directly proportional": one quantity is always a fixed multiple of the other.

Step 2. Isolate the constant. Rearranging $V = kT$ gives $k = V/T$ . Because $k$ depends only on the amount of gas and the (fixed) pressure — not on the particular values of $V$ and $T$ chosen — the ratio $V/T$ must come out to the same number no matter which moment you measure it at, as long as pressure and mass have not changed.

Step 3. Apply this to two different states of the same gas sample. Suppose the gas starts at volume $V_1$ and temperature $T_1$ , and is later found at volume $V_2$ and temperature $T_2$ , with the pressure held constant throughout and no gas added or removed. Since $V/T = k$ at every instant,

\frac{V_1}{T_1} = k = \frac{V_2}{T_2}

Step 4. Drop the middle term to get the working equation used in every Charles' Law problem:

\frac{V_1}{T_1} = \frac{V_2}{T_2}

This single relation is all that is needed to find any one of $V_1, T_1, V_2, T_2$ given the other three — provided both temperatures are converted to kelvin before substituting, since the proportionality $V \propto T$ (and therefore this derivation) is only true when $T$ is measured from absolute zero.

Worked example

A gas occupies 300 cm³ at 27 °C. Find its volume at 127 °C, with pressure held constant.

Step 1. Convert both temperatures to kelvin, since the equation only works with absolute temperature: $T_1 = 27 + 273 = 300\,\text{K}$ and $T_2 = 127 + 273 = 400\,\text{K}$ .

Step 2. Write down what is known and unknown: $V_1 = 300\,\text{cm}^3$ , $T_1 = 300\,\text{K}$ , $T_2 = 400\,\text{K}$ , and $V_2$ is to be found.

Step 3. Apply the Charles' Law equation and substitute the values:

\frac{V_1}{T_1} = \frac{V_2}{T_2} \quad\Rightarrow\quad \frac{300}{300} = \frac{V_2}{400}

Step 4. Solve for $V_2$ by cross-multiplying:

V_2 = \frac{300}{300} \times 400 = 400\,\text{cm}^3

Heating the gas from $300\,\text{K}$ to $400\,\text{K}$ — a rise from room temperature to well above boiling water — increases its volume from 300 cm³ to 400 cm³, exactly as expected since the temperature rose by a factor of $4/3$ and so did the volume. This proportional check ( $T_2/T_1 = V_2/V_1 = 4/3$ ) is a fast way to confirm an answer without redoing the algebra.

Reverse check — cooling to a target volume

Pass-1's second example runs the equation the other way: a gas of 546 cm³ at STP ( $T_1 = 273\,\text{K}$ ) is cooled at constant pressure until its volume falls to 500 cm³. Substituting into $V_1/T_1 = V_2/T_2$ gives $546/273 = 500/T_2$ , so $T_2 = (500 \times 273)/546 = 250\,\text{K}$ , which is $250 - 273 = -23\,^{\circ}\text{C}$ . Notice the pattern is the mirror image of the worked example above: volume fell, so temperature must also have fallen, and the ratio $V_2/V_1 = T_2/T_1$ still holds ( $500/546 = 250/273$ , both about $0.916$ ).

Common traps

Substituting Celsius values directly. $V_1/T_1 = V_2/T_2$ is only valid with kelvin; using 27 or 127 in place of 300 and 400 gives a nonsensical answer.
Forgetting the "constant pressure" condition. Charles' Law breaks down the moment pressure also changes — that situation needs the combined gas equation, not Charles' Law alone.
Believing the Celsius graph passes through the origin. It crosses the temperature axis at $-273\,^{\circ}\text{C}$ ; only the Kelvin-temperature graph passes through true zero on both axes.
Leaving the final answer in kelvin when the question specifically asks for a temperature "in °C" — always subtract 273 back at the end if the question demands Celsius.

Ask the tutor: Want to see how Charles' Law combines with Boyle's Law into the combined gas equation, or work through a few more temperature-to-volume problems together?

Why does a balloon in the sun get bigger but a sealed steel gas cylinder does not, even though both are heated

The difference is whether the container can change volume. A balloon has a flexible skin, so when the trapped air is heated its pressure would rise, but instead the skin stretches and the volume increases while the pressure stays close to atmospheric — a Charles' Law situation. A rigid steel cylinder cannot expand, so a fixed mass of gas inside it heated at constant volume instead shows a rise in pressure (a different gas law, Pressure Law, not covered by Charles' Law). The controlling condition, constant pressure versus constant volume, decides which law applies.

Do I always have to convert to kelvin even if both temperatures are given in Celsius

Yes, always. Charles' Law is a statement about absolute temperature, and the proportionality $V \propto T$ is only true when $T$ is measured from absolute zero at $-273\,^{\circ}\text{C}$ . Even if a question gives you two Celsius values and asks for an answer in Celsius, you must convert both to kelvin first, solve using $V_1/T_1 = V_2/T_2$ , and then convert your final kelvin answer back to Celsius if that is what was asked. Skipping the conversion is the single most common reason students lose marks on this topic.

How is Charles' Law different from Boyle's Law

Boyle's Law and Charles' Law both describe how a fixed mass of gas behaves, but they hold different quantities constant. Boyle's Law fixes the temperature and relates pressure and volume, which are inversely proportional ( $P_1V_1 = P_2V_2$ ). Charles' Law fixes the pressure and relates volume and temperature, which are directly proportional ( $V_1/T_1 = V_2/T_2$ ). A useful way to remember the difference is that Boyle's Law is about squeezing a gas at a steady temperature, while Charles' Law is about heating or cooling a gas that is free to expand or contract.

Charles' law

The core idea

Worked example

Reverse check — cooling to a target volume

Common traps

Before this

Where this leads