# 10.5: Universal Gas Law

- Page ID
- 2840

Compressed gases provide vital fuels for industry and for homes and farms in rural areas.

## Universal Gas Law

The combined gas law, PV∝T, is true for a particular sample of gas. If any gas is added or allowed to leak out, however, the relationship is lost. In order to get a relationship that is true for any sample of gas, it is necessary to incorporate a term for the amount of gas. From observations as simple as blowing up a balloon, it is clear that increasing the amount of gas increases the volume.

Because different gases have different weights per molecule, including a term for mass of gas does not produce a consistent equation. If, however, we include a term expressing the number of moles of gas rather than its mass, we can produce a constant proportionality. A mole is a unit representing the number of atoms present. The letter n is used to represent the moles of substance. Incorporating n into the equation yields PV∝nT. If we insert a letter, R, to represent the constant of proportionality, we get the normal form of the **universal gas law**, PV=nRT.

The unit term for n is always moles and T is always in Kelvin. The units for pressure and volume, however, may vary. The value of R depends on the units that are used for pressure and volume.

Pressure Units |
Volume Units |
Units for n_ |
Units for T_ |
Value of R_ |

atm | liters | moles | Kelvin | 0.0821 L•atm/mol•K |

atm | milliliters | moles | Kelvin | 82.1 mL•atm/mol•K |

Since the product of (liters)(atm) can be converted to joules, we also have a value for R where liters × atm have been converted to joules, R=8.314 J/mol⋅K. The two common values of the **universal gas law constant** **R **are 0.0821 L⋅atm/mol⋅K and 8.314 J/mol⋅K.

Most universal gas law problems are calculated at STP. STP stands for *standard temperature and pressure,* which is the most commonly calculated temperature and pressure value. STP is defined as 1.00 atm and 0°C, or 273 K.

## Examples

Example 10.5.1

Determine the volume of 1.00 mol of any gas at STP.

**Solution**

First isolate V from PV=nRT. Then plug in known values and solve.

V=nRT/P=(1.00 mol)(0.0821 L⋅atm/mol⋅K)(273 K)/(1.00 atm)=22.4 liters

For any gas at STP, one mole has a volume of 22.4 liters. This can be an extremely convenient conversion factor.

Example 10.5.2

A sample of oxygen gas occupies 10.0 liters at STP. How many moles of oxygen are in the container?

**Solution**

n=PV/RT=(1.00 atm)(10.0 L)/(0.0821 L⋅atm/mol⋅K)(273 K)=0.446 moles

Use the PLIX Interactive below to change the volume and temp erature of gas inside a piston and observe the how pressure, temperature and volume are related to each other in an ideal gas in a closed system:

Interactive Element

## Summary

- The universal gas law is PV=nRT, where P is pressure, V is volume, n is number of moles, R is the universal gas law constant, and T is the absolute temperature.
- The value of R varies depending on the units used for P and V. Two common values are 0.0821 L⋅atm/mol⋅K and R=8.314 J/mol⋅K.
- STP is standard temperature and pressure; 273 K and 1.00 atm.
- One mole of a gas at STP has a volume of 22.4 liters.

## Review

- The initial pressure in a helium gas cylinder is 30 atm. After many balloons have been blown up, the pressure in the cylinder has decreased to 6 atm while the volume and temperature remain the same. What fraction of the original amount of gas remains in the cylinder?
- Calculate the volume of 8.88 mol of helium gas at 20.0°C and 1.19 atm pressure.

## Explore More

Use this resource to answer the questions that follow.

- Why is it important to have values for R in kPa, atm, and mmHg?
- Why do the units of R include pressure, temperature, volume, and moles?

## Additional Resources

Real World Application: Hot Air Balloons

Videos:

Study Guide: Fluids Study Guide