- Correlate energy to motion of gas molecules.
- Correlate temperature to kinetic energy of gas molecules.
- Interpret pressure in terms of gas molecule motion.
- Describe effusion rate in terms of molecular motion.
- Estimate effusion rate and time by comparison

Temperature and pressure are macroscopic properties of gases. These
properties are related to molecular motion, which is a microscopic
phenomenon. **The kinetic theory of gases correlates between macroscopic
properties and microscopic phenomena**. Kinetics means the study of motion,
and in this case motions of gas molecules.

At the same temperature and volume, the same numbers of moles of all
gases exert the same pressure on the walls of their containers. This is
known as **Avogadros principle**. His theory implies that same numbers
of moles of gas have the same number of molecules.

Common sense tells us that the pressure is proportional to the average kinetic energy of all the gas molecules. Avogadros principle also implies that the kinetic energies of various gases are the same at the same temperature. The molecular masses are different from gas to gas, and if all gases have the same average kinetic energy, the average speed of a gas is unique.

Based on the above assumption or theory, Boltzmann (1844-1906) and Maxwell (1831-1879) extended the theory to imply that the average kinetic energy of a gas depends on its temperature.

They let* u* be the average or **root-mean-square speed** of a gas
whose molar mass is *M*. Since *N* is the Avogadro's number,
the average kinetic energy is (1/2) (*M/N) u*^{2}
or

Note thatM3R T3 K.E. = ---u^{2}= ---- = ---k T2N2 N 2

= (3

These formulas correlate temperature, pressure and kinetic energy of
molecules. The distribution of gas speed has been studied by Boltzmann
and Maxwell as well, but this is beyond the scope of this course.
However, you notice that at the same temperature, the average speed of
hydrogen gas, H_{2}, is 4 times more than that of oxygen,
O_{2} in order to have the same average kinetic energy.

For two gases, at the same temperature, with molecular masses
*M*_{1} and *M*_{2}, and average speeds*
u*_{1} and *u*_{2}, Boltzmann and Maxwell
theory implies the following relationship:

M_{1}u_{1}^{2}=M_{2}u_{2}^{2}. Thus,M_{1}u_{2}-- = (---)^{2 }M_{2}u_{1}

The consequence of the above property is that the effusion rate, the root mean square speed, and the most probable speed, are all inversely proportional to the square root (SQRT) of the molar mass. Simply formulated, the Graham's law of effusion is

d

The theories covered here enable you to make many predictions. Apply these theories to solve the following problems.

**
Example 1
**

*Solution*

Assume nitrogen behave as an ideal gas, then

= (

= 3742 J / mol (or 3.74 kJ/mol)

*Discussion*

At 300 K, any gas that behave like an ideal gas has the same energy per mol.

**
Example 2
**

*Solution*

Recall that

= (3

= (3*8.3145*310 / 0.002)

= 1966 m/s

= (3

= (3*8.3145*310)

= 87.9345/

Gas MolarH 2 1966 He 4 1390 Hu(root-mean-squar speed) mass /(m/s)_{2}28 525 O_{2}32 492 CO_{2}44 419

*Discussion*

Molar masses are 349 and 352 for ^{235}UF_{6} and
^{235}UF_{6} respectively. Using the method above, their
root-mean-square speeds are 149 and 148 m/s respectively.

The separation of these two isotopes of uranium was a necessity during the time of war for the US scientists. Gas diffusion was one of the methods employed for their separation.

**
Example 3
**

*Solution*

Since the effusion rates are

d

*Discussion*

The time required can be evaluated by

= 26.5 hr

© cchieh@uwaterloo.ca