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Molecular Velocities
Name: Tony A.
Status: student
Age: 12
Location: N/A
Country: N/A
Date: 12/2/2004
Question:
How fast do air molecule go in a normal environment? in
the freezer?
Replies:
Tony,
There's a really neat mathematical equation based on a theorem called
the "equipartition theorem" which states that the energy of a gas system
(equal to 1/2*mv^2) is equal to the temperature of the gas (equal to 3/2*kT).
If we rewrite this equation to solve for velocity we get:
sqrt(3*T*k/m) = v
where T is the temperature in Kelvin, k is the Boltzman constant = 1.3805*10^-
23 J/K and m is the mass of the gas particle.
If we assume that the average mass of air (since it is a mixture of different
gases) is 28.9 g/mol (or each gas particle is around 4.799*10^-26), and room-
temperature is 27C or 300K, we find that the average velocity of a single air
particle is around 500 m/s or 1100 miles per hour!
That is only the average. If you look up the idea called Maxwell's
distribution
of kinetic energy you will find that there is a small percentage of gas that
is travelling way faster than that.
Greg (Roberto Gregorius)
Not all molecules of a gas move at the same speed, there is a distribution
of speeds, so you have to think in terms of some sort of "average" speed.
The "kinetic theory of gases" (which you can search for more details) gives
several "averages": the most probably speed, the mean speed, or the root
mean square speed.
However these only differ be a constant whose value is approximately "1" --
the specific ratios are: 1.000..., 1.128, 1.225 respectively. In all cases the speed depends only on
the temperature expressed in kelvins and the molecular weight. The formula
for the average speed, which is written is:
= [(8/pi)*R*T/M]^1/2 where pi = 3.14159, R is the gas constant, T is the
temperature and M is the molar mass (that is the molecular weight). One has
to be careful to use a consistent set of units in the formula: For example
in SI units (joules, meters, kg/mol, sec): R = 8.314 J/mol. For oxygen: this
means: M=0.032kg/mol at 298 kelvins (approximately room temperature) = 25
C., the value of = 444 m/sec.
This is about 993 miles/hour. This depends upon temperature like ~ (T)^1/2.
So at temperatures in the range of 0 C to 100 C the average speed does not
vary very much.
Vince Calder
Tony-
This kind of thing can be calculated. You will learn how in high
school chemistry or physics.
The speed of sound in gasses is set by the thermal speed of the molecules.
So your air molecules are going something like 1000 feet per second, 740
miles/hour, 330 meters/second.
The temperature (above absolute zero) sets the average kinetic energy of
everything that wiggles or moves from heat:
E = Kb x T
(Kb is Boltzmann's thermal constant: kB = 1.380658 x 10-23 Joules/Kelvin. A Joule is energy: 1 watt for 1 second. )
The kinetic energy is the Square of the velocity of the molecule:
E = 1/2 M V^2 = (1/2) x M x (V x V).
(M for mass of 1 average
air molecule 4.8 x 10^-27 kilograms; V in meters/sec; E in Joules )
So the speed of molecules and sound goes as the square-root of temperature:
V = Sqrt[ 2 X E / M ]
= Sqrt[ 2 x (Kb x T) / M ]
= Sqrt[ something x T ]
= Sqrt[something] x Sqrt[T]
V = something2 x Sqrt[T].
("something" = 2 x Kb / M and "something2" = Sqrt[something])
Make sure you use the Kelvin temperature, the temperature above absolute zero.
Absolute zero is the temperature with truly zero thermal energy, where gas
molecules have no velocity.
(Having no velocity, they'd stick to each other for any little reason, so
I guess they would no longer be gasses anyway.)
Absolute zero is -273 C, so our room temperature (say it's 25C)
is really 273 + 25 = 298 degrees Kelvin.
So convert degrees F to degrees C, then add 273.
If your freezer is 20 degrees F, that would be:
( 20 degreeF -32 degreeF ) / 1.8 = -6.67 degrees C
-6.67 degreeC + 273 = 266.3 degrees Kelvin
V(freezer) / V(room) =
= Sqrt[ T(freezer) ] / Sqrt[ T(room) ]
= Sqrt[ T(freezer) / T(room) ]
= Sqrt[ 266 / 298 ]
= Sqrt[ 0.89 ]
= 0.95
= 95%
= 100% - 5%
Molecules only go about 5% slower in your freezer.
Dry ice, or somewhere just above liquid-nitrogen temperature, would be
much better for having slower air molecules.
But they would still be above 50% of room-temperature speed.
Below 50%, they get stuck together, and you get liquid air rather than a gas.
Helium atoms are the least sticky atom there is, so they can go about 8
times slower than at room temperature before clumping together.
Jim Swenson
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