Graphs make the ideal gas law visual. By holding two variables constant and plotting the other two, you can recognise curve shapes, interpret slopes, and predict how changing conditions shifts a gas’s measured pressure or volume.

Core idea: graphs come from the ideal gas law

Most AP Chemistry gas graphs are “slices” of ideal-gas behaviour where two state variables are held constant. The graph’s shape communicates whether the relationship is direct (linear) or inverse (hyperbolic), and whether a straight-line replot is possible (e.g., using $1/V$).

On any gas graph, read the prompt carefully for what is constant: typical constraints are “at constant $T$ and $n$” or “at constant $P$.”

This equation is the source of the proportionalities used to interpret each graph.

P–V graphs (constant T and n)

Shape and interpretation

With constant $T$ and $n$, rearrange to $P=\dfrac{nRT}{V}$, so pressure is inversely proportional to volume.

• A plot of $P$ vs. $V$ is a downward-curving hyperbola.

Boyle’s law graphical representation: pressure decreases nonlinearly as volume increases for a fixed amount of gas at constant temperature. The curve visually encodes the inverse proportionality $P\propto 1/V$, which is why equal fractional changes in $V$ do not produce equal absolute changes in $P$. Source

• The curve gets steeper at small $V$ because changing $V$ strongly affects collision frequency with container walls.

Linearising the relationship

AP questions often use a straight-line version to test proportional reasoning.

• Plotting $P$ vs. $1/V$ gives a straight line through the origin for ideal behaviour.

• The line’s steepness increases when $T$ or $n$ is larger (because $nRT$ is larger).

Families of curves

Multiple isotherms on the same axes show:

• Higher $T$ isotherm: lies above a lower $T$ curve (at the same $V$, higher $T$ gives higher $P$).

V–T graphs (constant P and n)

Why Kelvin matters

With constant $P$ and $n$, $V=\dfrac{nR}{P}T$, so volume is directly proportional to absolute temperature.

• A plot of $V$ vs. $T$ (in K) is a straight line.

A volume–temperature plot illustrating Charles’s law at constant pressure and moles: the data fall on a straight line when temperature is measured in kelvins. The line extrapolates to zero volume at 0 K, highlighting why absolute temperature is required for a direct proportionality $V\propto T$. Source

• Using Celsius would shift the intercept and hide the proportionality; AP expects Kelvin for direct proportional relationships.

Slope meaning

For $V$ vs. $T$ at constant $P$:

• Larger $n$ makes the slope larger (more moles expand more per kelvin).

• Larger $P$ makes the slope smaller (pressure resists expansion).

P–T graphs (constant V and n)

With constant $V$ and $n$, $P=\dfrac{nR}{V}T$, so pressure is directly proportional to absolute temperature.

• A plot of $P$ vs. $T$ (K) is linear.

• Increasing $n$ increases the slope; increasing $V$ decreases the slope.

Graphs involving n (amount of gas)

When $T$ and $P$ are constant, $V=\dfrac{RT}{P}n$ so:

• $V$ vs. $n$ is linear (doubling moles doubles volume).

When $T$ and $V$ are constant, $P=\dfrac{RT}{V}n$ so:

• $P$ vs. $n$ is linear (more moles means more wall collisions and higher pressure).

How to use these graphs to predict changes

Graphical representations help you connect a change in conditions to a shift in a line or curve:

• Direct relationships ($V$–$T$, $P$–$T$, $V$–$n$, $P$–$n$): look for straight lines and compare slopes.

• Inverse relationships ($P$–$V$): look for curvature or use a $1/V$ axis to compare lines.