The **combined gas law** is an ideal gas law that combines Boyle’s law, Charles’s law, and Gay-Lussac’s law. It states the the ratio between the pressure-volume product and absolute temperature of a gas is a constant. Pressure, volume, and temperature are allowed to change, but the amount of gas (number of moles) remains unchanged. Basically, the combined gas law is the ideal gas law, except that it’s missing Avogadro’s law. The combined gas law doesn’t have an official discoverer, so it doesn’t have a name.

- Combined gas law: PV/T = k
- Boyle’s law: PV = k
- Charles’s law: V/T = k
- Gay-Lussac’s law: P/T = k

### Combined Gas Law Formula

The basic formula for the combined gas law related pressure (P), volume (V), and absolute temperature (T):

**PV/T = k**

The constant k is a true constant, so long as the number of moles of a gas remains the same. If the amount of gas varies, then k changes.

The practical formula for the combined gas law gives “before and after” conditions of a gas:

**P _{1}V_{1} / T_{1} = P_{2}V_{2} / T_{2}**

Rearranging the variables gives:

**P _{1}V_{1} T_{2} = P_{2}V_{2} T_{1}**

Any units of pressure and volume are fine, but temperature must be absolute. In other words, convert Fahrenheit and Celsius temperatures to Kelvin.

### How the Combined Gas Law Applies to Everyday Life

The combined gas law has practical applications in everyday life. It applies whenever the amount of gas remains constant, but pressure, volume, and temperature change. For example, the law predicts the behavior of cloud formation, refrigerators, and air conditioners. It’s also used in other thermodynamics and fluid mechanics calculations.

Because the combined gas law is an ideal gas law, it only approximates the behavior of real gases. The law becomes less accurate at high pressures and temperatures.

### Combined Gas Law Example Problem

Find the volume of a gas at 760.0 mm Hg pressure and 273 K when 2.00 liters is collected at 745.0 mm Hg and 25.0 °C.

First convert 25.0 °C to the Kelvin scale. This gives you 298 Kelvin.

Next, plug the values into the combined gas law formula. The most common mistake students make is mixing up which numbers go together. Writing down what you’re given helps avoid this error:

P_{1} = 745.0 mm Hg

V_{1} = 2.00 L

T_{1} = 298 K

P_{2} = 760.0 mm Hg

V_{2} = x (the unknown you’re solving for)

T_{2} = 273 K

Arrange the formula to solve for the unknown:

P_{1}V_{1} / T_{1} = P_{2}V_{2} / T_{2}

P_{1}V_{1}T_{2} = P_{2}V_{2}T_{1}

V_{2} = (P_{1}V_{1}T_{2}) / (P_{2}T_{1})

Plug in the numbers:

V_{2} = (745.0 mm Hg · 2.00 L · 273 K) / (760 mm Hg · 298 K)

V_{2} = 1.796 L

V_{2} = 1.80 L

### References

- Castka, Joseph F.; Metcalfe, H. Clark; Davis, Raymond E.; Williams, John E. (2002).
*Modern Chemistry*. Holt, Rinehart and Winston. ISBN 978-0-03-056537-3. - Fullick, P. (1994).
*Physics*. Heinemann. ISBN 978-0-435-57078-1. - Raff, Lionel M. (2001)
*Principles of Physical Chemistry*(1st ed.). Pearson College Div. ISBN: 978-0130278050.