Combined Gas Law

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:

combined-gas-law

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 V1, and we are also told that the container is at STP, standard temperature and pressure. Standard temperature is 273.15 K, which is T1, and standard pressure is 1 atm or 101.3 kPa, which is P1. 150 kPa is P2 and T2 is 26 ºC. We want to find out the volume V2 when the pressure is 150 kPa and temperature is 26 ºC.

V1 = 2.54 L

T1 = 273.15 K

P1 = 1 atm = 101.3 kPa

V2 = ?

T2 = 26 ºC

P2 = 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:

combined-299

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

The Combined Gas Law is:

combined-gas-law

To isolate the variable V2, multiply each side by T2/P2:

combined-gas-law-isolate

We get:

combined-gas-law-v2

Plugging in our values for P1, V1, T2, P2 and T1:

combined-gas-law-19L

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?

V1 = 8.40 L

When the pressure is reduced to one-half its original pressure, P2 = ½P1.

Since its temperature is doubled, T2 = 2T1.

We are asked to find the new volume, V2.

V1 = 8.40 L

T1 = T1

P1 = P1

V2 = ?

T2 = 2T1

P2 = ½P1

The combined gas law is:

combined-gas-law

To solve for V2, we multiply each side by T2/P2:

combined-V2

Our equation becomes:

combined-V22

Plugging in our values for P1, V1, T2, P2 and T1, we get:

combined-v2-prob

Notice that P1 and T1 cancel out:

combined-prob2

And we get:

combined-v2-336L

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 V1. That makes 13 L V2. Since 542 torr is associated with V1 of 21 L, 542 torr is P1. We want to find the new pressure P2 in torr.

V1 = 21 L

P1 = 542 T

V2 = 13 L

P2 = ?

The combined gas law is:

combined-gas-law

Since temperature is not mentioned at all in this problem, we can assume it is constant before and after. Since temperature is constant, T1 = T2:

combined-boyles

Our equation becomes Boyle’s Law:

boyles-law

Solving for P2:

boyles-law-p2

Plugging in our values for P1, V1 and V2:

boyles-law-880t

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 V1. The original temperature T1 is 89.0 °C. The problem asks for the new volume of oxygen V2 when the temperature drops to the new temperature T2 = 0.00 °C.

V1 = 150. L

T1 = 89.0 °C

V2 = ?

T2 = 0.00 °C

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

combined-gas-law-temp-conversions

V1 = 150. L

T1 = 89.0 °C = 362.15 K

V2 = ?

T2 = 0.00 °C = 273.15 K

The Combined Gas Law is:

combined-gas-law

Since there is no mention of pressure in this problem, it is assumed to be constant, the same before and after. Since P1 = P2:

combined-gas-law-charles

The combined gas law becomes Charles’s Law:

charles-law

Solving for V2, the equation becomes:

charles-law-v2

Plugging in our values for V1, T2, and T1:

combined-gas-law-113L

The new volume of oxygen is 113 L.