19.8:
Maxwell-Boltzmann Distribution: Problem Solving
Consider an oxygen gas in a container having a molar mass of 0.0320 kg/mol at room temperature. Determine the ratio of the number of molecules with a speed of around 400 m/s to the number of molecules with a speed of around 200 m/s, the root-mean-square speed, and the most probable speed of oxygen molecules at room temperature.
To solve the problem, identify the known and unknown quantities.
In differential form, the number of molecules close to a speed is the product of the probability distribution function of the speed times a small interval of the speed.
The ratio can be calculated by dividing the number of molecules close to the individual speeds.
Substitute the terms in the Maxwell-Boltzmann distribution function for gas molecules. The ratio of the number of molecules with individual speeds can be determined.
Recall the root-mean-square speed equation and the most probable speed equation for gas molecules. By substituting the known values, the root-mean-square speed and the most probable speed of the gas can be determined.
19.8:
Maxwell-Boltzmann Distribution: Problem Solving
Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
Since N is dimensionless and the unit of f(v) is seconds per meter, the equation can be conveniently modified into a differential form:
Consider a sample of nitrogen gas in a cylinder with a molar mass of 28.0 g/mol at a room temperature of 27 °C. Determine the ratio of the number of molecules with a speed very close to 300 m/s to the number of molecules with a speed very close to 100 m/s.
To solve the problem, examine the situation, and identify known and unknown quantities.
Second, convert all the known values into proper SI units. For example, convert the molecular weight to kilograms (kg) and the temperature to kelvin (K). Recall the distribution function for the velocity equation. Lastly, substitute the known values into the equation to determine the unknown quantity.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version], pages 94–95. Retrieved from https://openstax.org/books/university-physics-volume-2/pages/2-4-distribution-of-molecular-speeds