The Kinetic Theory of Gases
A Maple Exercise
The kinetic theory of gases is one of the cornerstones of physical chemistry. It provides a model that allows the calculation of many dynamic properties of a perfect gas. Knowledge of molecular speeds and their distribution functions is useful in understanding the rates of gas phase reactions.
Theory
The kinetic theory of gases was developed from a model that incorporated the following features:
a) the gas consists of large numbers of particles in continual random motion,
b) the size of the particles is negligible in comparison to the average distance traveled between collisions, and
c) the particles exert no intermolecular forces on one another and thus their collisions are perfectly elastic.
The predictions of the kinetic theory are found to agree well with the actual behavior of real gases at normal pressures and temperatures well above their boiling points.
According to the kinetic theory, the temperature of a gas is a measure of the average translational kinetic energy of the gas, and is thus also a measure of the average speed, <c>, of the gas particles. Common sense (and experimental results) tells us that not all gas particles move with the average speed — some move faster than average and some slower. A distribution function provides the fraction of molecules with speeds between c and c + dc. The distribution function for perfect gases is called the Maxwell-Boltzmann distribution function and is given below:
F(c) dc = 4
π m
2
πkT
3/2
c2 exp − mc2
2kT
dc [1]
where m is the molecular mass, k is the Boltzmann constant, and T is the Kelvin temperature. Note especially the dependence of this function on mass and temperature. This equation can be plotted as a function of c to see how the fraction of molecules with a given speed changes with speed or temperature. The above equation can also be used for several purposes. The maximum of the distribution function provides the most probable speed, c *. Differentiating Equation [1] with respect to c and solving for c when the derivative equals zero one obtains the most probable speed. c* = 2kT
m
1/ 2
The fraction of particles with speeds between c 1 and c 2 can be obtained by integrating equation [1] between c 1 and c 2.
One of the most powerful uses of distribution functions is calculating averages. For any distribution function F(x), the average of any property which depends on x (such as ξ) is given byξ = ξ F(x)∫ dx [3] Using this, the average speed can be determined by evaluating c = c F(c)∫ dc [4] where the limits of integration range from c=0 to c=∞. The resulting integral is c = 8kTπm
1/2
In a similar manner, the root-mean-square speed, c rms, is determined by evaluating c2 1/ 2 ≡ crms = c2 F(c)dc∫( )1/ 2
The integral evaluates to the following simple expression. crms = 3kT
m
1/ 2
The root-mean-square speed is particularly useful since the average translational kinetic energy EK is given by
EK = 1
2 m c2 = 1
2 mcrms
2
Combining equations [7] and [8] yields
EK = 3
2 kT
the average translational energy per particle for a perfect gas. This result agrees well with experimental values for monatomic gases. Note that the average kinetic energy is not a function of the mass of the particles involved.
The kinetic theory also provides information regarding molecular collisions. The number of collisions experienced by a single particle per second per unit volume is given by
z A = 2
πd2 c N
V = 2
πd2 c p
kT
where d is the molecular diameter and N the number of particles per volume V. The quantity π d2 is often called the collision cross section and is given the symbol
σ. The total number of collisions between particles per second per unit volume is given by Z AA = πd2 c N2 2
Calculations
1. Determine the Maxwell-Boltzmann distribution function for hydrogen gas and oxygen gas at 298 K (plot data for both on same graph). Remember, these gases are diatomic and check you units!
2. Determine the Maxwell-Boltzmann distribution function for oxygen at 298 K and 500 K (plot data for both on same graph)
4. Using your value for the most probable speed at 298 K for oxygen determined in part 3, determine the fraction of molecules at 298 K and at 500 K which have speeds greater than this speed. Assuming this is the minimum speed required for molecules to be travelling in order to react in some specific reaction, estimate the relative reaction rates at the two temperatures. Make sure you use the same speed for both curves.
