7.1: The Kinetic-Molecular Theory
- Page ID
- 395209
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- State the postulates of the kinetic-molecular theory
- Use this theory’s postulates to explain the gas laws
Steam, water, and ice are often considered three distinct substances because their physical properties are so different. We know that whether water freezes or boils is temperature dependent and this leads to an interesting question: what is temperature? What, exactly, does temperature measure. Temperature is a way to represent the average speed of particles. A lower temperature means molecules are moving slower, while a higher temperature means the molecules are moving faster.
The temperature at which particles have no movement, defined as 0 K or -273.15 °C.
The kinetic-molecular theory (KMT) is a simple molecular model that effectively explains the physical properties of matter using the motion of the molecules. This theory, focusing on gases, is based on the following five postulates described here. (Note: The term “molecule” will be used to refer to the individual chemical species that compose the gas, although some gases are composed of atomic species, for example, the noble gases.)
- Gases are composed of molecules that are in continuous motion, travelling in straight lines and changing direction only when they collide with other molecules or with the walls of a container.
- The molecules composing the gas are negligibly small compared to the distances between them.
- The pressure exerted by a gas in a container results from collisions between the gas molecules and the container walls.
- Gas molecules exert no attractive or repulsive forces on each other or the container walls; therefore, their collisions are elastic (do not involve a loss of energy).
- The average kinetic energy of the gas molecules is proportional to the kelvin temperature of the gas.
The test of the KMT and its postulates is its ability to explain and describe the behavior of a gas. The various gas laws can be derived from the assumptions of the KMT, which have led chemists to believe that the assumptions of the theory accurately represent the properties of gas molecules. We will first look at the individual gas laws (Boyle’s, Charles’s, Amontons’s, Avogadro’s, and Dalton’s laws) conceptually to see how the KMT explains them. Then, we will more carefully consider the relationships between molecular masses, speeds, and kinetic energies with temperature, and explain Graham’s law.
The Kinetic-Molecular Theory Explains the Behavior of Gases, Part I
Gases consist of rapidly moving gas molecules and, if inside a container, will hit a unit area of the wall per unit of time. The impacts will create pressure in the container. We see that the KMT conceptually explains the behavior of a gas as follows:
- Amontons’s law. If the temperature is increased, the average speed and kinetic energy of the gas molecules increase. If the volume is held constant, the increased speed of the gas molecules results in more frequent and more forceful collisions with the walls of the container, therefore increasing the pressure (Figure \(\PageIndex{1a}\)).
- Charles’s law. If the temperature of a gas is increased, a constant pressure may be maintained only if the volume occupied by the gas increases. This will result in greater average distances traveled by the molecules to reach the container walls, as well as increased wall surface area. These conditions will decrease both the frequency of molecule-wall collisions and the number of collisions per unit area, the combined effects of which balance the effect of increased collision forces due to the greater kinetic energy at the higher temperature.
- Boyle’s law. If the gas volume is decreased, the container wall area decreases and the molecule-wall collision frequency increases, both of which increase the pressure exerted by the gas (Figure \(\PageIndex{1b}\)).
- Avogadro’s law. At constant pressure and temperature, the frequency and force of molecule-wall collisions are constant. Under such conditions, increasing the number of gaseous molecules will require a proportional increase in the container volume in order to yield a decrease in the number of collisions per unit area to compensate for the increased frequency of collisions (Figure \(\PageIndex{1c}\)).
- Dalton’s Law. Because of the large distances between them, the molecules of one gas in a mixture bombard the container walls with the same frequency whether other gases are present or not, and the total pressure of a gas mixture equals the sum of the (partial) pressures of the individual gases.
Molecular Velocities and Kinetic Energy
The previous discussion showed that the KMT qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws. To do this, we must first look at velocities and kinetic energies of gas molecules, and the temperature of a gas sample.
In a gas sample, individual molecules have widely varying speeds; however, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are constant. This molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed (Figure \(\PageIndex{2}\)).
The kinetic energy (KE) of a particle of mass (m) and speed (v) is given by:
\[\ce{KE}=\dfrac{1}{2}mv^2 \nonumber \]
Expressing mass in kilograms and speed in meters per second will yield energy values in units of joules (J = kg m2 s–2). To deal with a large number of gas molecules, we use averages for both speed and kinetic energy. In the KMT, the root mean square velocity of a particle, urms, is defined as the square root of the average of the squares of the velocities with n = the number of particles:
\[u_\ce{rms}=\sqrt{\overline{u^2}}=\sqrt{\dfrac{v^2_1+v^2_2+v^2_3+v^2_4+…}{n}} \nonumber \]
The average kinetic energy, KEavg, is then equal to:
\[\mathrm{KE_{avg}}=\dfrac{1}{2}mv^2_\ce{rms} \nonumber \]
The KEavg of a collection of gas molecules is also directly proportional to the temperature of the gas and may be described by the equation:
\[\mathrm{KE_{avg}}\propto T \nonumber \]
where we can then substitute kinetic energy and see
\[\mathrm{mv^2_{avg}}\propto T \nonumber \]
The average velocity of the molecules is inverse to the mass of the molecule, assuming temperature is constant. Similarly, we can see that velocity increases with temperature, assuming mass is constant.
Order the following gases at the same temperature from least relative velocity to greatest relative velocity: oxygen gas, chlorine gas, hydrogen gas.
Solution
Determine the formula mass of each molecule. Oxygen gas is 32 amu, chlorine gas is 71 amu, and hydrogen gas is 1.0 amu. Mass is inversely proportional to mass, thus the order from least to greatest will be the opposite of the masses.
\[\text{Least}\ \rightarrow\ \text{Greatest}\]
\[\text{Mass}\]
\[\text{hydrogen}\ \rightarrow\ \text{oxygen}\ \rightarrow\ \text{chlorine}\]
\[\text{Relative Velocity}\]
\[\text{chlorine}\ \rightarrow\ \text{oxygen}\ \rightarrow\ \text{hydrogen}\]
Order the following gases at the same temperature from least relative velocity to greatest relative velocity: bromine gas, fluorine gas, and helium gas.
- Answer
-
\(\text{bromine}\ \rightarrow\ \text{fluorine}\ \rightarrow\ \text{helium}\)
If the temperature of a gas increases, its KEavg increases, more molecules have higher speeds and fewer molecules have lower speeds, and the distribution shifts toward higher speeds overall, that is, to the right. If temperature decreases, KEavg decreases, more molecules have lower speeds and fewer molecules have higher speeds, and the distribution shifts toward lower speeds overall, that is, to the left. This behavior is illustrated for nitrogen gas in Figure \(\PageIndex{3}\).
At a given temperature, all gases have the same KEavg for their molecules. Gases composed of lighter molecules have more high-speed particles and a higher urms, with a speed distribution that peaks at relatively higher velocities. Gases consisting of heavier molecules have more low-speed particles, a lower urms, and a speed distribution that peaks at relatively lower velocities. This trend is demonstrated by the data for a series of noble gases shown in Figure \(\PageIndex{4}\).
The Kinetic-Molecular Theory Explains the Behavior of Gases, Part II
The molecules of a gas are in rapid motion and the molecules themselves are small. The average distance between the molecules of a gas is large compared to the size of the molecules. As a consequence, gas molecules can move past each other easily and diffuse at relatively fast rates.
The rate of effusion of a gas depends directly on the (average) speed of its molecules:
\[\textrm{effusion rate} ∝ v_\ce{rms} \nonumber \]
Using this relation, and the equation relating molecular speed to mass, Graham’s law may be easily derived as shown here:
\[u_\ce{rms}=\sqrt{\dfrac{3RT}{m}} \nonumber \]
\[m=\dfrac{3RT}{u^2_\ce{rms}}=\dfrac{3RT}{\overline{u}^2} \nonumber \]
\[\mathrm{\dfrac{effusion\: rate\: A}{effusion\: rate\: B}}=\dfrac{u_\mathrm{rms\:A}}{u_\mathrm{rms\:B}}=\dfrac{\sqrt{\dfrac{3RT}{m_\ce{A}}}}{\sqrt{\dfrac{3RT}{m_\ce{B}}}}=\sqrt{\dfrac{m_\ce{B}}{m_\ce{A}}} \nonumber \]
The ratio of the rates of effusion is thus derived to be inversely proportional to the ratio of the square roots of their masses.
Kinetic-Molecular Theory of Gases: https://youtu.be/9f83XAYfXAg
Summary
The kinetic molecular theory is a simple but very effective model that effectively explains ideal gas behavior. The theory assumes that gases consist of widely separated molecules of negligible volume that are in constant motion, colliding elastically with one another and the walls of their container with average velocities determined by their absolute temperatures. The individual molecules of a gas exhibit a range of velocities, the distribution of these velocities being dependent on the temperature of the gas and the mass of its molecules.
Key Equations
- \(\mathrm{KE_{avg}}\propto T \nonumber \)
- \(\mathrm{mv^2_{avg}}\propto T \nonumber \)
Summary
- kinetic molecular theory
- theory based on simple principles and assumptions that effectively explains the properties of states of matter