Root Mean Square Velocity, Effusion, and Diffusion

From the KMT, Kelvin temperature indicates the average kinetic
energy of the gas particles
PV
2
= RT = (KE)avg
n
3
• 2/3 KE comes from the application of velocity, momentum, force, and pressure
when deriving an expression for pressure
 See Appendix 2 in your book for a complete mathematical explanation!

Thus,
(KE)avg
 ALL
3
= RT
2
gases have the same average kinetic
energy at the same temperature!
 This
mathematical relationship is very
important because it shows that with
higher temperature comes greater
motion of the gas particles
• HEAT ‘EM UP – SPEED ‘EM UP!

Look at the graph at
right
• How do the number
of gaseous
molecules with a
given velocity
change with
increasing
temperature?
 By drawing a vertical
line from the peak of
each bell curve to the
x-axis, the AVERAGE
velocity of the sample
is derived

Average velocity of a specific gas molecule at a
specific temperature is also called its root mean
square velocity (μrms)
• Can be calculated using Maxwell’s equation:
μ2
= μrms =
3RT
MM
• Where:
 R is “energy R” = 8.314 J/K∙mol
 T = temperature in Kelvin
 MM = molar mass of a single gas particle in KILOGRAMS per
mole!
• μrms has units of m/s!

This equation is important because it shows that molar mass is
inversely proportional to velocity
• Massive particles move slowly
• Light particles move quickly

But remember - ALL gases have the same average
kinetic energy at the same temperature!
 Calculate
the root mean square velocity
for the atoms in a sample of helium gas at
25°C.
 We
have seen that the postulates of the
KMT, when combined with appropriate
physical principles, produce an equation
that successfully fits the experimentally
observed behavior of gases
 There are 2 further tests of this model:
• Effusion
• Diffusion
If we could monitor the
path of a single
molecule, it would be
very erratic
 The average distance a
particle travels
between collisions is
called the mean free
path

• At sea level, mean free
path is about 6 x 10-6 cm
 Waaaaayyyyy small!

A man named Thomas
Graham studied the
passage of a gas
through a tiny orifice
into an evacuated
chamber
• Called effusion

Only the gas molecules
that hit the small hole
would escape through
it
• Therefore, the higher the
root mean square
velocity of a gas particle,
the more likelihood it
would hit the hole

Thomas Graham experimentally showed that the rate of
effusion of a gas is inversely proportional to the square root
of the mass of its particles


Called the rate of effusion
 It measures the speed at which the gas is transferred into the
chamber
Stated in another way, the relative rates of effusion of two
gases at the same temperature and pressure are given by
the inverse ratio of the square roots of the masses of the gas
particles
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑒𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑓𝑜𝑟 𝑔𝑎𝑠 1 𝜇𝑟𝑚𝑠 𝑓𝑜𝑟 𝑔𝑎𝑠 1
=
=
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑒𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑓𝑜𝑟 𝑔𝑎𝑠 2 𝜇𝑟𝑚𝑠 𝑓𝑜𝑟 𝑔𝑎𝑠 2
 M1 and M2 = molar masses of the gases in g/mol
3𝑅𝑇
𝑀1
3𝑅𝑇
𝑀2
=
𝑀2
𝑀1
 Calculate
the ratio of the effusion rates of
hydrogen gas (H2) and uranium
hexafluoride (UF6), a gas used in the
enrichment process to produce fuel for
nuclear reactors

Diffusion describes the spread of a gas through
space
• Faster for light gas molecules than for heavier gas
molecules

Diffusion is significantly slower than rms speed
• Consider someone opening a perfume bottle
 It takes a while to detect the odor but rms speed at 25°C is
~1150 mi/hr


Diffusion is slowed with the mixing of gases
Quite complicated to describe theoretically
because so many collisions occur when gases
mix
• However, diffusion is slowed by gas molecules colliding
with each other