12.6: The Combined Gas Law
- Understand mathematical relationship between pressure, volume, and temperature give the amount of gas is constant.
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 T 1 = T 2 : then \(P_{1} V_{1}=P_{2} V_{2}\), this is the Boyle's law.
2. If P 1 = P 2 : then \(\dfrac{V_{1}}{T_{1}}=\dfrac{V_{2}}{T_{2}}\), this is the Charles's law.
3. If V 1 = V 2 : 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.
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.