Themes > Science > Chemistry > Inorganic Chemistry > More Information About Gas Laws > Gas Index > Kinetic Theory of Gases: Gas particle speed

The kinetic theory of gases models gases by assuming that they are a bunch of rapidly moving atoms or molecules that have an average translational kinetic energy proportional to their temperature. The kinetic energy of a particle is given by E = 0.5*m*v2, where m is the mass of the particle and v it's velocity. If we use the average speed u rather than a single particle speed v, then E = 0.5*m*u2. Since we know this is proportional to the temperature, we can show
0.5*m*u2 = cT

The proportionality constant c can be computed: it turns out to be c = 3R/2Na, where R is the ideal gas constant and Na is Avagadro's number. Since the molar mass of a substance (M) is just Na* m, we can show

u = (3RT / M)1/2
This gives us a relationship between the average speed of a gas molecule u and the temperature T. Note that the speed is proportional to the square root of the temperature (Molecules move faster when heated) and inversely proportional to the molar mass. (Heavy molecules move more slowly than light ones.)

When using the above equation, you must be careful to keep the units consistant. For most gas problems, we use the value

R = 0.0821 L*atm/mol*K
This will not give an answer in useful units for this problem. Use instead the value of R in units of (g*m2) / (s2*mol*K)
R = 8.314*103 (g*m2) / (s2*mol*K)

Example: In a sample of oxygen gas at 250C, how fast are the gas molecules moving?

Solution: The molar mass of oxygen gas, O2 is 32.0 g/mole. Simply plug into the above equation, making sure your units are correct.

u = (3RT / M)1/2
u = (3* 8.314*103 (g*m2/s2*mol*K) * 298K/ 32 g/mole)1/2
u = (2.32*105 m2/s2)1/2
u = 482 m/s
Note that the units came out in m/s, a velocity as we need. This is about 900 miles/hour.


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