# 13.4 Kinetic theory: atomic and molecular explanation of pressure  (Page 2/6)

 Page 2 / 6

We can get the average kinetic energy of a molecule, $\frac{1}{2}{\text{mv}}^{2}$ , from the right-hand side of the equation by canceling $N$ and multiplying by 3/2. This calculation produces the result that the average kinetic energy of a molecule is directly related to absolute temperature.

$\overline{\text{KE}}=\frac{1}{2}m\overline{{v}^{2}}=\frac{3}{2}\text{kT}$

The average translational kinetic energy of a molecule, $\overline{\text{KE}}$ , is called thermal energy     . The equation $\overline{\text{KE}}=\frac{1}{2}m\overline{{v}^{2}}=\frac{3}{2}\text{kT}$ is a molecular interpretation of temperature, and it has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.

It is sometimes useful to rearrange $\overline{\text{KE}}=\frac{1}{2}m\overline{{v}^{2}}=\frac{3}{2}\text{kT}$ , and solve for the average speed of molecules in a gas in terms of temperature,

$\sqrt{\overline{{v}^{2}}}={v}_{\text{rms}}=\sqrt{\frac{3\text{kT}}{m}},$

where ${v}_{\text{rms}}$ stands for root-mean-square (rms) speed.

## Calculating kinetic energy and speed of a gas molecule

(a) What is the average kinetic energy of a gas molecule at $\text{20}\text{.}0\text{º}\text{C}$ (room temperature)? (b) Find the rms speed of a nitrogen molecule $\left({\text{N}}_{2}\right)$ at this temperature.

Strategy for (a)

The known in the equation for the average kinetic energy is the temperature.

$\overline{\text{KE}}=\frac{1}{2}m\overline{{v}^{2}}=\frac{3}{2}\text{kT}$

Before substituting values into this equation, we must convert the given temperature to kelvins. This conversion gives $T=\left(\text{20}\text{.}0+\text{273}\right)\phantom{\rule{0.25em}{0ex}}\text{K = 293}\phantom{\rule{0.25em}{0ex}}\text{K}.$

Solution for (a)

The temperature alone is sufficient to find the average translational kinetic energy. Substituting the temperature into the translational kinetic energy equation gives

$\overline{\text{KE}}=\frac{3}{2}\text{kT}=\frac{3}{2}\left(1\text{.}\text{38}×{\text{10}}^{-\text{23}}\phantom{\rule{0.25em}{0ex}}\text{J/K}\right)\left(\text{293}\phantom{\rule{0.25em}{0ex}}\text{K}\right)=6\text{.}\text{07}×{\text{10}}^{-\text{21}}\phantom{\rule{0.25em}{0ex}}\text{J}\text{.}$

Strategy for (b)

Finding the rms speed of a nitrogen molecule involves a straightforward calculation using the equation

$\sqrt{\overline{{v}^{2}}}={v}_{\text{rms}}=\sqrt{\frac{3\text{kT}}{m}},$

but we must first find the mass of a nitrogen molecule. Using the molecular mass of nitrogen ${\text{N}}_{2}$ from the periodic table,

$m=\frac{2\left(\text{14}\text{.}\text{0067}\right)×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{kg/mol}}{6\text{.}\text{02}×{\text{10}}^{\text{23}}\phantom{\rule{0.25em}{0ex}}{\text{mol}}^{-1}}=4\text{.}\text{65}×{\text{10}}^{-\text{26}}\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}$

Solution for (b)

Substituting this mass and the value for $k$ into the equation for ${v}_{\text{rms}}$ yields

${v}_{\text{rms}}=\sqrt{\frac{3\text{kT}}{m}}=\sqrt{\frac{3\left(1\text{.}\text{38}×{\text{10}}^{–\text{23}}\phantom{\rule{0.25em}{0ex}}\text{J/K}\right)\left(\text{293 K}\right)}{4\text{.}\text{65}×{\text{10}}^{\text{–26}}\phantom{\rule{0.25em}{0ex}}\text{kg}}}=\text{511}\phantom{\rule{0.25em}{0ex}}\text{m/s}\text{.}$

Discussion

Note that the average kinetic energy of the molecule is independent of the type of molecule. The average translational kinetic energy depends only on absolute temperature. The kinetic energy is very small compared to macroscopic energies, so that we do not feel when an air molecule is hitting our skin. The rms velocity of the nitrogen molecule is surprisingly large. These large molecular velocities do not yield macroscopic movement of air, since the molecules move in all directions with equal likelihood. The mean free path (the distance a molecule can move on average between collisions) of molecules in air is very small, and so the molecules move rapidly but do not get very far in a second. The high value for rms speed is reflected in the speed of sound, however, which is about 340 m/s at room temperature. The faster the rms speed of air molecules, the faster that sound vibrations can be transferred through the air. The speed of sound increases with temperature and is greater in gases with small molecular masses, such as helium. (See [link] .)

Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
how would I work this problem
Alexia
how can you have not an integer number of protons? If, on the other hand it supposed to be 1e12, then 1.6e-19C/proton • 1e12 protons=1.6e-7 C is the charge of the protons in the speck, so the difference between this and 5e-9C is made up by electrons
Igor
what is angular velocity
Why does earth exert only a tiny downward pull?
hello
Islam
Why is light bright?
an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
thanks so much. i undersooth well
what is physics
is the study of matter in relation to energy
Kintu
a submersible pump is dropped a borehole and hits the level of water at the bottom of the borehole 5 seconds later.determine the level of water in the borehole
what is power?
power P = Work done per second W/ t. It means the more power, the stronger machine
Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
did you solve?
Shii
1.75cm
Ridwan
my name is Abu m.konnek I am a student of a electrical engineer and I want you to help me
Abu
the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
Kings
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
hi
Godfred
Godfred
If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
What was the previous far point of a patient who had laser correction that reduced the power of her eye by 7.00 D, producing a normal distant vision power of 50.0 D for her?
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?
Jaydie
29/20 ? maybes
Ju
In what ways does physics affect the society both positively or negatively