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Combined Gas Equation

Key Concepts

For a fixed quantity of gas in moles, Boyle's Law and Charles' Law can be combined into one equation, the combined gas equation:

PV
T = k (a constant)

OR
P₁V₁
T₁ = P₂V₂
T₂

where:
P₁ = initial pressure P₂ = final pressure

V₁ = initial volume V₂ = final volume

T₁ = initial temperature (K) T₂ = final temperature (K)

Examples

Pressure Calculation

A quantity of gas occupied a volume of 1.0L at 1atm pressure and 80^oC.
What pressure is required to compress the gas to a volume of 500mL at 40^oC?

Organize the data and make all the units consistent:
P₁ = 1atm P₂ = ?atm

V₁ = 1.0L V₂ = 500mL = 500 x 10^-3L

T₁ = 80^oC = 80 + 273 = 353K T₂ = 80^oC = 40 + 273 = 313K

Re-arrange the combined gas equation:
P₂
= P₁V₁T₂
T₁V₂

Substitute the knowns into the equation:
P₂
= 1 x 1.0 x 313
353 x 500 x 10^-3

Solve for P₂:

P₂ = 1.77atm

Volume Calculation

A quantity of gas has volume of 24.5L at 101.3kPa and 298K.
What volume will this gas occupy if it is cooled to 0^oC at a pressure of 2atm?

Organize the data and make all the units consistent:
P₁ = 101.3kPa = 1atm P₂ = 2atm

V₁ = 24.5L V₂ = ?L

T₁ = 298K T₂ = 0^oC = 0 + 273 = 273K

Re-arrange the combined gas equation:
V₂
= P₁V₁T₂
T₁P₂

Substitute the knowns into the equation:
V₂
= 1 x24.5 x 273
298 x 2

Solve for V₂:

V₂ = 11.22L

Temperature Calculation

A quantity of gas has volume of 2.5L at 760mm Hg and 20^oC.
If the gas is compressed to a volume of 750mL by a pressure of 2atm what is its temperature in Kelvin?

Organize the data and make all the units consistent:
P₁ = 760mm Hg = 1atm P₂ = 2atm

V₁ = 2.5L V₂ = 750mL = 750 x 10^-3L

T₁ = 20^oC = 293K T₂ = ?K

Re-arrange the combined gas equation:
T₂
= P₂V₂T₁
P₁V₁

Substitute the knowns into the equation:
T₂
= 2 x 750 x 10^-3 x 293
1 x 2.5

Solve for T₂:

T₂ = 175.8K

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