Thermal physics

Gas laws

Explore how pressure, volume, and temperature of a fixed amount of gas relate in isothermal, isobaric, and isochoric processes.

Gas laws describe how an equilibrium state changes when one variable is held fixed. Always use absolute temperature $T$ in kelvins; Celsius values cannot be substituted into ratios.

pV=nRT,\qquad pV=\text{const (isothermal)},\quad \frac{V}{T}=\text{const (isobaric)},\quad \frac{p}{T}=\text{const (isochoric)}

Definition: Three basic processes

For a fixed amount of gas: an isothermal process keeps $T$ fixed, so $pV$ is constant; an isobaric process keeps $p$ fixed, so $V/T$ is constant; an isochoric process keeps $V$ fixed, so $p/T$ is constant. These relations assume near-ideal gas behavior and equilibrium states.

Choose a process and adjust the sliders to follow the gas state. Each graph is meaningful only when the process constraints are respected.

Reading the graphs

In the $p$ – $V$ plane, an ideal-gas isotherm is the hyperbola $p=nRT/V$ . Isochores are vertical and isobars horizontal. A path represents states traversed, not the rate at which time passes along it.

Example: Isothermal compression

A gas has volume $4.0$ L at pressure $100$ kPa and is compressed isothermally to $2.5$ L. Find its final pressure.

Solution

Boyle's law gives $p_1V_1=p_2V_2$ , so $p_2=100(4.0/2.5)=160$ kPa.

Quick check

Each gas law describes a particular transformation of the same fixed amount of gas. Isothermal means absolute temperature stays constant, so $pV$ stays constant: compressing the gas reduces volume and raises pressure. Isobaric keeps pressure constant, making volume proportional to kelvin temperature. Isochoric keeps volume constant, so pressure is proportional to kelvin temperature. A graph helps identify the constraint: an isotherm in the $p$ – $V$ plane is a hyperbola, while an isobar is horizontal. The combined relation $p_1V_1/T_1=p_2V_2/T_2$ packages these changes for a fixed amount of gas; it does not mean every variable can vary independently. Before applying it, identify what is held fixed, convert Celsius to kelvins when taking ratios, and use consistent units for both states. These laws follow from the ideal-gas equation of state.

For example, compressing gas isothermally from 4.0 L at 100 kPa to 2.5 L gives $p_2=p_1V_1/V_2=160$ kPa. Pressure rises because the same amount of gas is confined to a smaller volume; Celsius temperatures must not be used in state ratios.

An ideal gas is heated at constant volume. What happens to its absolute pressure?

For an isothermal process of a fixed amount of ideal gas, what remains constant?

References

Charles Kittel, Herbert Kroemer (1980). Thermal Physics
Halliday, Resnick, Walker (2018). Fundamentals of Physics