Chemistry » Gases » The Kinetic-Molecular Theory

Molecular Velocities and Kinetic Energy

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 (see the figure below).

A graph is shown. The horizontal axis is labeled, “Velocity v ( m divided by s ).” This axis is marked by increments of 20 beginning at 0 and extending up to 120. The vertical axis is labeled, “Fraction of molecules.” A positively or right-skewed curve is shown in red which begins at the origin and approaches the horizontal axis around 120 m per s. At the peak of the curve, a point is indicated with a black dot and is labeled, “v subscript p.” A vertical dashed line extends from this point to the horizontal axis at which point the intersection is labeled, “v subscript p.” Slightly to the right of the peak a second black dot is placed on the curve. This point is labeled, “v subscript r m s.” A vertical dashed line extends from this point to the horizontal axis at which point the intersection is labeled, “v subscript r m s.” The label, “O subscript 2 at T equals 300 K” appears in the open space to the right of the curve.

The molecular speed distribution for oxygen gas at 300 K is shown here. Very few molecules move at either very low or very high speeds. The number of molecules with intermediate speeds increases rapidly up to a maximum, which is the most probable speed, then drops off rapidly. Note that the most probable speed, νp, is a little less than 400 m/s, while the root mean square speed, urms, is closer to 500 m/s.

The kinetic energy (KE) of a particle of mass (m) and speed (u) is given by:

\(\text{KE}=\phantom{\rule{0.2em}{0ex}}\cfrac{1}{2}\phantom{\rule{0.2em}{0ex}}m{u}^{2}\)

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:

\({u}_{\text{r}\text{m}\text{s}}=\sqrt{\overline{{u}^{2}}}=\sqrt{\cfrac{{u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}+{u}_{4}^{2}+\dots }{n}}\)

The average kinetic energy for a mole of particles, KEavg, is then equal to:

\({\text{KE}}_{\text{avg}}=\phantom{\rule{0.2em}{0ex}}\cfrac{1}{2}{Mu}_{\text{rms}}^{2}\)

The KEavg of a mole of gas molecules is also directly proportional to the temperature of the gas and may be described by the equation:

\({\text{KE}}_{\text{avg}}=\phantom{\rule{0.2em}{0ex}}\cfrac{3}{2}\phantom{\rule{0.2em}{0ex}}RT\)

where R is the gas constant and T is the kelvin temperature. When used in this equation, the appropriate form of the gas constant is 8.314 J/mol⋅K (8.314 kg m2s–2mol–1K–1). These two separate equations for KEavg may be combined and rearranged to yield a relation between molecular speed and temperature:

\(\cfrac{1}{2}{Mu}_{\text{rms}}^{2}=\phantom{\rule{0.2em}{0ex}}\cfrac{3}{2}\phantom{\rule{0.2em}{0ex}}RT\)

\({u}_{\text{rms}}=\sqrt{\cfrac{3RT}{M}}\)

Example: Calculation of urms

Calculate the root-mean-square velocity for a nitrogen molecule at 30 °C.

Solution

Convert the temperature into Kelvin:

\(30\phantom{\rule{0.2em}{0ex}}\text{°C}+273=\text{303 K}\)

Determine the molar mass of nitrogen in kilograms:

\(\frac{28.0\phantom{\rule{0.2em}{0ex}}\require{cancel}\cancel{\text{g}}}{\text{1 mol}}\phantom{\rule{0.4em}{0ex}}×\phantom{\rule{0.4em}{0ex}}\frac{\text{1 kg}}{1000\phantom{\rule{0.2em}{0ex}} \require{cancel}\cancel{\text{g}}}\phantom{\rule{0.2em}{0ex}}=0.028\phantom{\rule{0.2em}{0ex}}\text{kg/mol}\)

Replace the variables and constants in the root-mean-square velocity equation, replacing Joules with the equivalent kg m2s–2:

\({u}_{\text{rms}}=\sqrt{\frac{3RT}{m}}\)

\({u}_{\text{r}\text{m}\text{s}}=\phantom{\rule{0.2em}{0ex}}\sqrt{\frac{3\left(8.314\phantom{\rule{0.2em}{0ex}}\text{J/mol K}\right)\left(\text{303 K}\right)}{\left(0.028\phantom{\rule{0.2em}{0ex}}\text{kg/mol}\right)}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\sqrt{2.70\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}{\text{s}}^{-2}}\phantom{\rule{0.2em}{0ex}}=519\phantom{\rule{0.2em}{0ex}}\text{m/s}\)

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 the figure below.

A graph with four positively or right-skewed curves of varying heights is shown. The horizontal axis is labeled, “Velocity v ( m divided by s ).” This axis is marked by increments of 500 beginning at 0 and extending up to 1500. The vertical axis is labeled, “Fraction of molecules.” The label, “N subscript 2,” appears in the open space in the upper right area of the graph. The tallest and narrowest of these curves is labeled, “100 K.” Its right end appears to touch the horizontal axis around 700 m per s. It is followed by a slightly wider curve which is labeled, “200 K,” that is about three quarters of the height of the initial curve. Its right end appears to touch the horizontal axis around 850 m per s. The third curve is significantly wider and only about half the height of the initial curve. It is labeled, “500 K.” Its right end appears to touch the horizontal axis around 1450 m per s. The final curve is only about one third the height of the initial curve. It is much wider than the others, so much so that its right end has not yet reached the horizontal axis. This curve is labeled, “1000 K.”

The molecular speed distribution for nitrogen gas (N2) shifts to the right and flattens as the temperature increases; it shifts to the left and heightens as the temperature decreases.

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 the figure below.

A graph is shown with four positively or right-skewed curves of varying heights. The horizontal axis is labeled, “Velocity v ( m divided by s ).” This axis is marked by increments of 500 beginning at 0 and extending up to 3000. The vertical axis is labeled, “Fraction of molecules.” The tallest and narrowest of these curves is labeled, “X e.” Its right end appears to touch the horizontal axis around 600 m per s. It is followed by a slightly wider curve which is labeled, “A r,” that is about half the height of the initial curve. Its right end appears to touch the horizontal axis around 900 m per s. The third curve is significantly wider and just over a third of the height of the initial curve. It is labeled, “N e.” Its right end appears to touch the horizontal axis around 1200 m per s. The final curve is only about one fourth the height of the initial curve. It is much wider than the others, so much so that its right reaches the horizontal axis around 2500 m per s. This curve is labeled, “H e.”

Molecular velocity is directly related to molecular mass. At a given temperature, lighter molecules move faster on average than heavier molecules.

Note:

The gas simulator may be used to examine the effect of temperature on molecular velocities. Examine the simulator’s “energy histograms” (molecular speed distributions) and “species information” (which gives average speed values) for molecules of different masses at various temperatures.

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