7.5: The combined gas law
The laws relating to pressure \(P\), volume \(V\), and temperature \(T\) for a constant amount \(n\) of a gas are the following:
- If \(n\) and \(T\) are constant: \(P_{1} V_{1}=P_{2} V_{2}\), that is Boyle's law.
- If \(P\) and \(n\) are constant: \(\dfrac{V_{1}}{T_{1}}=\dfrac{V_{2}}{T_{2}}\), that is Charles's law.
- If \(\mathrm{V}\) and \(\mathrm{n}\) are constant: \(\dfrac{P_{1}}{T_{1}}=\dfrac{P_{2}}{T_{2}}\), that is Gay Lussac's law.
All three relationships are combined in the following law.
If \(\mathrm{n}\) is constant: \[\dfrac{P_{1} V_{1}}{T_{1}}=\dfrac{P_{2} V_{2}}{T_{2}}\nonumber\], that is the combined gas 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 \(25^{\circ} \mathrm{C}\) and \(750 \mathrm{~mm} \mathrm{Hg}\). What is the volume of the balloon when it ascends to an altitude where the temperature is \(-40{ }^{\circ} \mathrm{C}\) 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):
\[\begin{array}{lll}
\mathrm{P}_{1}=750 \mathrm{~mm} \mathrm{Hg}, & \mathrm{V}_{1}=212 \mathrm{~L}, & \mathrm{~T}_{1}=25^{\circ} \mathrm{C}+273.15=298.15 \mathrm{~K} \\
\mathrm{P}_{2}=540 \mathrm{~mm} \mathrm{Hg}, & \mathrm{V}_{2}=? & \mathrm{~T}_{2}=-40^{\circ} \mathrm{C}+273.15=233.15 \mathrm{~K}
\end{array}\nonumber\]
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:
\[V_{2}=\dfrac{750 \cancel{\mathrm{~mm} \mathrm{Hg}} \times 212 \mathrm{~L} \times 233.15 \cancel{\mathrm{~K}}}{298.15 \cancel{\mathrm{~K}} \times 540 \cancel{\mathrm{~mm} \mathrm{Hg}}}=230 \mathrm{~L}. \nonumber\]