7.5: The combined gas law
- Page ID
- 371603
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[V_{2}=\dfrac{750 \cancel{\mathrm{~mm} \mathrm{Hg}} \times 212 \mathrm{~L} \times 233.15 \cancel{\mathrm{~K}}}{298.15 \cancel{\mathrm{~K}} \times 540 \cancel{\mathrm{~mm} \mathrm{Hg}}}=230 \mathrm{~L}. \nonumber\]