12.6: The Combined Gas Law

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Learning Objectives

Understand mathematical relationship between pressure, volume, and temperature give the amount of gas is constant.

Combined gas law

The laws relating to pressure \(P\), volume \(V\), and temperature \(T\) for a constant amount of a gas are the following:

If amount of gas is constant: \[\dfrac{P_{1} V_{1}}{T_{1}}=\dfrac{P_{2} V_{2}}{T_{2}}\nonumber\]

This is a combination of Boyle's law, Charles law and the Gay-Lussac's law.

1. If T₁ = T₂: then \(P_{1} V_{1}=P_{2} V_{2}\), this is the Boyle's law.

2. If P₁ = P₂: then \(\dfrac{V_{1}}{T_{1}}=\dfrac{V_{2}}{T_{2}}\), this is the Charles's law.

3. If V₁ = V₂: then \(\dfrac{P_{1}}{T_{1}}=\dfrac{P_{2}}{T_{2}}\), this is the Gay Lussac's law.

The combined gas law allows calculating the effect of varying two parameters on the third.

Example \(\PageIndex{1}\)

A weather balloon contains \(212 \mathrm{~L}\) of helium at \(25.0^{\circ} \mathrm{C}\) and \(750. \mathrm{~mm} \mathrm{Hg}\). What is the volume of the balloon when it ascends to an altitude where the temperature is \(-40.0{ }^{\circ} \mathrm{C}\) and \(540. \mathrm{~mm} \mathrm{Hg}\), assuming the quantity of gas remains the same?

Solution

Given and desired parameters (temperatures must be converted to Kelvin scale):

\[\begin{array}{lll}
\mathrm{P}_{1}=750. \mathrm{~mm} \mathrm{Hg}, & \mathrm{V}_{1}=212 \mathrm{~L}, & \mathrm{~T}_{1}=25.0^{\circ} \mathrm{C}+273.15=298.2 \mathrm{~K} \\
\mathrm{P}_{2}=540. \mathrm{~mm} \mathrm{Hg}, & \mathrm{V}_{2}=? & \mathrm{~T}_{2}=-40.0^{\circ} \mathrm{C}+273.15=233.2 \mathrm{~K}
\end{array}\nonumber\]

Formula:

\[\dfrac{P_{1} V_{1}}{T_{1}}=\dfrac{P_{2} V_{2}}{T_{2}}, \nonumber\]

rearrange the formula to isolate the desired parameter:

\[V_{2}=\dfrac{P_{1} V_{1} T_{2}}{T_{1} P_{2}}. \nonumber\]

Calculations:

\[V_{2}=\dfrac{750. \cancel{\mathrm{~mm} \mathrm{Hg}} \times 212 \mathrm{~L} \times 233.2 \cancel{\mathrm{~K}}}{298.2 \cancel{\mathrm{~K}} \times 540. \cancel{\mathrm{~mm} \mathrm{Hg}}}=230. \mathrm{~L}. \nonumber\]

Summary

The combined gas law is a combination of Boyle's, Charles', and the Gay-Lussac's law.