Combined Gas Law:

Mathematically, this can be represented as:

Substituting in variables, the formula is:

Because the formula is equal to a constant, it is possible to solve for a change in volume, temperature, or pressure using a proportion:

Explanation and Discussion:

The Combined Gas Law is a combination of Boyle's and Charles' Laws. The Combined Gas Law describes the relationship between pressure, volume, and temperature. For example, if the pressure increased, wither the volume would decrease or the temperature would increase.

The Combined Gas Law can be used to solve any Boyle's or Charles' Law problem. If one of the variables remains constant, disregard that variable by leaving it out of your equation or setting it to the same value on either side of the equals sign.

Calculations with the Combined Gas Law: Let's try a problem with the Combined Gas Law. For example, the pressure and temperature of a gas are changed to STP (101.3 kPa/0°C) from 22.0°C and 30.8 kPa. What will be the new volume if the original volume was 205 mL?

Set-up

First, we must convert degrees Celcius to Kelvins. This means that our original temperature was 295 K and our target temperature is 273 K.

Estimate and Answer

It seems that the pressure increases, but the temperature decreases. Because of their inverse relationship, the increased pressure will decrease the volume. Because of their direct relationship, the decreased temperature will decrease the volume, also. If we consider the amount that the pressure increases, it more than triples. The temperature decreases only slightly. Therefore, it is safe to assume that the new volume will be less than a third of its original volume, or less than 68.2 mL (205 x 0.3333 = 68.2).

Plug values into the formula

From the problem, we know that, for the original state of the gas, the conditions were:
Volume (V) = 205 mL
Temperature (T) = 295 K
Pressure (P) = 30.8 kPa

We also know the target temperature and pressure:

Temperature₁ (T₁) = 273 K K
Pressure₁ (P₁) = 101.3 kPa

The unkown is V₁. Our equation is:

30.8 kPa x 205 mL / 295 K = 101.3 kPa x V₁ / 273 K

Manipulating the equation, we get:

V₁ = 30.8 kPa x 205 mL / 295 K x 273 K / 101.3 kPa

V₁ = 57.68 mL

However, because our problem only was to three digits of precision, the answer is:

V₁ = 57.7 mL

Check

To check our answer, we must compare it to our earlier estimate. We estimated the value to be less than 68.2 mL. Because 57.7 mL is less than 68.2 mL, but not incredible much so (like 20 - 30 mL), we can assume the answer to be correct.

Continued Study:
Because the Combined Gas Law is derived from Charles' Law and Boyle's Law, you can find more information at those pages. You can also test yourself.

Brown, Theodore L., H. Eugene LeMay, Jr. and Bruce E. Burston, Chemistry: The Central Science, Englewood Cliffs, NJ: Prentice Hall, Inc., 1994

Dorin, Henry, Peter E. Demmin, and Dorothy L. Gabel. Prentice Hall Chemistry: The Study of Matter, Needham, Massachusetts and Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1989.

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