# Combined Gas Law

The three gas laws Charles’s Law, Boyle’s Law and Gay-Lussac’s Law can be combined into one combined gas law relating pressure, volume and temperature of a gas. The combined gas law is:

We’ll learn how to use this equation by considering some sample problems.

**Sample Problem**

Carbon dioxide occupies a 2.54 L container at STP. What will be the volume when the pressure is 150 kPa and 26 ºC?

The volume 2.54 L is V_{1}, and we are also told that the container is at STP, standard temperature and pressure. Standard temperature is 273.15 K, which is T_{1}, and standard pressure is 1 atm or 101.3 kPa, which is P_{1}. 150 kPa is P_{2} and T_{2} is 26 ºC. We want to find out the volume V_{2} when the pressure is 150 kPa and temperature is 26 ºC.

V_{1} = 2.54 L

T_{1} = 273.15 K

P_{1} = 1 atm = 101.3 kPa

V_{2} = ?

T_{2} = 26 ºC

P_{2} = 150 kPa

Before we plug these values into the Combined Gas Law, we need to convert all temperatures to Kelvin. To convert 26 ºC to Kelvin:

Click here to learn more about converting between celsius and Kelvin temperature units.

The Combined Gas Law is:

To isolate the variable V_{2}, multiply each side by T_{2}/P_{2}:

We get:

Plugging in our values for P_{1}, V_{1}, T_{2}, P_{2} and T_{1}:

*The volume will be 1.9 L.*

**Sample Problem**

The pressure of 8.40 L of nitrogen gas is decreased to one-half its original pressure, and its temperature is doubled. What is the new volume?

V_{1} = 8.40 L

When the pressure is reduced to one-half its original pressure, P_{2} = ½P_{1}.

Since its temperature is doubled, T_{2} = 2T_{1}.

We are asked to find the new volume, V_{2}.

V_{1} = 8.40 L

T_{1} = T_{1}

P_{1} = P_{1}

V_{2} = ?

T_{2} = 2T_{1}

P_{2} = ½P_{1}

The combined gas law is:

To solve for V_{2}, we multiply each side by T_{2}/P_{2}:

Our equation becomes:

Plugging in our values for P_{1}, V_{1}, T_{2}, P_{2} and T_{1}, we get:

Notice that P_{1} and T_{1} cancel out:

And we get:

*The new volume is 33.6 L.*

The individual Gas Laws can be derived from the Combined Gas Law

Boyle’s Law

Charles’s Law

Gay-Lussac’s Law

See the next sample problem to find out how.

**Sample Problem**

Helium in a sealed syringe is compressed to a volume of 13 L. Its original volume was 21 L at 542 torr. Find the new pressure in torr.

The original volume of the gas is 21 L, so we’ll let that be known as V_{1}. That makes 13 L V_{2}. Since 542 torr is associated with V_{1} of 21 L, 542 torr is P_{1}. We want to find the new pressure P_{2} in torr.

V_{1} = 21 L

P_{1} = 542 T

V_{2} = 13 L

P_{2} = ?

The combined gas law is:

Since temperature is not mentioned at all in this problem, we can assume it is constant before and after. Since temperature is constant, T_{1} = T_{2}:

Our equation becomes Boyle’s Law:

Solving for P_{2}:

Plugging in our values for P_{1}, V_{1} and V_{2}:

*The new pressure is 880 Torr.*

**Sample Problem**

A sample of oxygen occupies a volume of 150. L at 89.0 °C. What will be the volume of oxygen when the temperature drops to 0.00 °C?

The original volume of the oxygen is 150. L which is V_{1}. The original temperature T_{1} is 89.0 °C. The problem asks for the new volume of oxygen V_{2} when the temperature drops to the new temperature T_{2} = 0.00 °C.

V_{1} = 150. L

T_{1} = 89.0 °C

V_{2} = ?

T_{2} = 0.00 °C

First we need to convert temperature in celsius to temperature in Kelvin.

V_{1} = 150. L

T_{1} = 89.0 °C = 362.15 K

V_{2} = ?

T_{2} = 0.00 °C = 273.15 K

The Combined Gas Law is:

Since there is no mention of pressure in this problem, it is assumed to be constant, the same before and after. Since P_{1} = P_{2}:

The combined gas law becomes Charles’s Law:

Solving for V_{2}, the equation becomes:

Plugging in our values for V_{1}, T_{2}, and T_{1}:

*The new volume of oxygen is 113 L.*