Pressure Proportional to Temperature? The Gas Law

Pressure is proportional to absolute temperature for a fixed amount of gas in a fixed-volume container, provided the gas behaves approximately ideally. This relationship, called Gay-Lussac’s law, lets you estimate how pressure changes when temperature changes; it does not apply if volume or the amount of gas also changes.

Formula

At constant volume and with a fixed amount of gas, pressure divided by absolute temperature is constant:

P₁ / T₁ = P₂ / T₂, or P₂ = P₁ × T₂ / T₁.

• P₁ is the initial absolute pressure, in kilopascals (kPa) in this calculator.
• T₁ is the initial absolute temperature, in kelvins (K).
• P₂ is the final absolute pressure, in kPa.
• T₂ is the final absolute temperature, in K.

The temperature ratio is dimensionless. The calculator assumes the initial and final pressure use the same units and that the gas amount and container volume remain unchanged. Pressure must be absolute, measured from a vacuum, rather than gauge pressure, measured relative to the surrounding atmosphere.

Worked Example

Using the calculator’s defaults, the gas starts at 100 kPa and 300 K, then reaches 330 K. The container is assumed to remain rigid and sealed.

1. Write the relation: P₂ = P₁ × T₂ / T₁.
2. Substitute the values: P₂ = 100 kPa × 330 K / 300 K.
3. Calculate the temperature ratio: 330 / 300 = 1.1.
4. Multiply: P₂ = 100 × 1.1 = 110.00 kPa.

The temperature has risen by 10%, so the absolute pressure rises by 10% under these conditions. The result is an estimate from the ideal-gas relationship, not a safety rating for a container.

Why Pressure Tracks Temperature

Gas particles move in constant, random motion and collide with the walls of their container. Pressure is the force of those collisions per unit area. Heating a gas increases the particles’ average kinetic energy; in a fixed space, they move faster and transfer momentum to the walls more vigorously. The resulting pressure increase is captured by the proportional relationship between pressure and absolute temperature.

The broader ideal gas law is PV = nRT, where P is absolute pressure, V is volume, n is the amount of gas in moles, R is the gas constant, and T is absolute temperature. If V and n stay fixed, the equation reduces to P/T = constant. This is why the conditions matter: changing container volume or allowing gas to enter or escape changes the result.

Use Absolute Temperature and Pressure

A common mistake is to use Celsius or Fahrenheit values directly in a ratio. The zero points on those scales are not absolute zero, so their numerical values are not proportional to molecular kinetic energy. Convert temperatures first:

• Kelvin = Celsius + 273.15.
• Kelvin = (Fahrenheit + 459.67) × 5/9.
• Rankine = Fahrenheit + 459.67; Rankine and kelvin are both absolute scales.

For example, heating a rigid sealed gas from 20°C to 40°C means comparing 293.15 K with 313.15 K, not comparing 20 with 40. The pressure factor is about 1.068, an increase of roughly 6.8%, not 100%.

Pressure units also need attention. The ratio equation works with any consistent absolute-pressure unit, such as pascals, kilopascals, bars, or pounds per square inch absolute (psia). Gauge pressure (psig) excludes atmospheric pressure, so convert it before using the equation: absolute pressure equals gauge pressure plus local atmospheric pressure. Near sea level, atmospheric pressure is about 101.3 kPa, or 14.7 psi, but actual atmospheric pressure varies with altitude and weather.

When the Relationship Applies—and Its Limits

Use this relationship for a sealed, rigid vessel whose gas remains in a single phase and is not so dense or cold that non-ideal behavior becomes important. It is useful for understanding pressure changes in a fixed-volume gas sample and for making first-order estimates. It does not by itself predict the pressure in a flexible container, a leaky system, or a process where the gas can change phase.

• Volume changes: If a piston moves or a balloon expands, the gas may gain volume rather than develop the full pressure increase. Use the full ideal gas law or a more suitable process model.
• Gas amount changes: A vent, leak, or gas-generating reaction changes n, so pressure is not determined by temperature alone.
• Non-ideal gas behavior: At high pressures or near condensation, real gas particles do not follow the ideal model exactly. A real-gas equation or measured properties may be needed.
• Temperature is not uniform: A changing temperature may take time to spread through a vessel. The simple calculation assumes one representative equilibrium temperature.

Do not use this equation as a substitute for a vessel’s pressure rating, engineering review, or operating instructions. Heating a sealed container can create dangerous pressure; never heat a container unless it is designed and approved for that use.

Related Gas-Law Comparisons

Several familiar gas laws are special cases of PV = nRT. Gay-Lussac’s law describes pressure versus temperature when volume and gas amount are fixed. Charles’s law describes volume versus absolute temperature when pressure and gas amount are fixed. Boyle’s law describes pressure versus volume when temperature and gas amount are fixed. The combined gas law accounts for changes in pressure, volume, and temperature for a fixed amount of gas:

P₁V₁ / T₁ = P₂V₂ / T₂.

All temperatures in these relations must be absolute, and pressures must be absolute. The ideal-gas assumption still applies. If the amount of gas changes, the more general equation must include n as well.

Frequently Asked Questions

Is pressure directly proportional to temperature?
Yes, if the gas amount and volume are constant and temperature is measured on an absolute scale. Under those conditions, doubling the kelvin temperature doubles the absolute pressure.

Does gas pressure increase when temperature increases?
In a fixed-volume, sealed gas sample, yes. Faster-moving particles collide more forcefully with the container walls. In a flexible or vented system, the volume or amount of gas may change instead.

Can I use Celsius in the pressure-temperature formula?
No. Convert Celsius to kelvins before calculating a temperature ratio. Celsius values can be used to find a temperature difference, but not as the absolute temperatures in P₁/T₁ = P₂/T₂.

Can I use gauge pressure?
Not directly. Convert gauge pressure to absolute pressure by adding the local atmospheric pressure, calculate the final absolute pressure, then subtract atmospheric pressure if you need the answer as gauge pressure.

What happens to pressure if the container expands?
Pressure may rise less, remain similar, or even fall, depending on how much the volume changes and on the process. The fixed-volume proportional rule no longer applies; use a relation that includes both pressure and volume.

Related

• Record Video on Mac: Camera, Screen, and Audio
• National AI Compute Cloud GPUs vs On-Premises Hardware
• Integrate Google Calendar With Notion: Options and Setup

in p1 | Initial pressure | kPa | 100 | 0
in t1 | Initial temperature | K | 300 | 0
in t2 | Final temperature | K | 330 | 0
out p2 | Final pressure | kPa | p1 * t2 / t1 | 2
out ratio | Pressure change factor | - | t2 / t1 | 3