12.3: The Pressure-Temperature Law
- Page ID
- 435127
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Understand the relationship between pressure of a gas and temperature.
The properties of a gas can be related to each other under certain conditions. The properties are pressure (P), volume (V), temperature (T, in kelvins), and amount of material expressed in moles (n). A sample of gas cannot have any random values for these properties. Instead, only certain values, dictated by some simple mathematical relationships, will occur.
The first simple relationship, referred to as a gas law, is between the pressure of a gas and its temperature. If the amount of gas in a sample and its volume are kept constant, then as the temperature of a gas is increased, the pressure of the gas increases proportionately. Consider a gas in a cylinder with a piston in Figure \(\PageIndex{1}\). Increasing temperature increases the average kinetic energy (KE) and the average velocity of the gas molecules resulting in more frequent and more forceful collisions which result in increased gas pressure applied on the piston or the walls of the gas container.
Gay-Lussac’s law states that the pressure of a gas is directly proportional to the absolute temperature provided the volume and amount of gas are not changed.
The mathematical forms of Gay-Lussac’s law are the following.
\[P\propto{T}\nonumber\], or \[P=\mathrm{k}T\nonumber\], or \[\frac{P}{T}=\mathrm{k}, \nonumber\]
where \(k\) is a constant, \(P\) is pressure, and \(T\) is the temperature (in kelvin scale) of the gas. Since Since \(\frac{P}{T}\) is a constant, it implies that
\[\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}=\mathrm{k}, \nonumber\]
where \(P_1\) is the initial pressure, \(T_1\) is the initial temperature in Kelvin, \(P_2\) is the final pressure, and \(T_2\) is the final temperature in Kelvin, provided the amount of gas and volume do not change.
The pressure of an oxygen tank containing 15.0 L oxygen is 965 torr at 55.0 oC. What will be the pressure when the tank is cooled to 16.0 oC.
Solution
First, convert the temperatures to the Kelvin scale before applying gas laws.
Given: T1 = 55.0 oC + 273.15 = 328.2 K, T2 = 16.0 oC + 273.15 = 289.2 K, P1 = 965 torr, P2 = ?
Formula:
\[\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}, \nonumber\]
rearrange the formula to isolate the desired variable:
\[P_{2}=\frac{P_{1} T_{2}}{T_{1}} \nonumber\]
Plug in the values in the rearranged formula and calculate:
\[P_{2}=\frac{965 \mathrm{~torr} \times 289.2\mathrm{~K}}{328.2 \mathrm{~K}}=850. \mathrm{~torr} \nonumber\]
Applications
Oxygen Cylinders
The relationship between temperature and pressure must be kept in mind while using pressurized O2 (g) cylinders in the hospital. When oxygen is administered to a patient the pressure of the gas in the tank decreases. As per Gay-Lussac's law the temperature of the tank will decrease too. The surrounding will be cooler too. Be sure to provide an extra blanket to the your patients to keep them comfortable.
Car Tires
It is important to keep your car tires inflated to the correct pressure to maximize gas mileage. The recommended inflation pressure is based on the measurements made when the tire is cold. As you drive your car, your tire heats up. For every 10 oF increase in temperature, the tire pressure increases by 1 psi.
Summary
The pressure of a gas is directly proportional to the temperature in the Kelvin scale provided the volume and amount of gas is not changed.