"Equilibrium Constants & Probability"
The goal of this project is to construct probability model to calculate the equilibrium
constant of hydrogen and deuterium exchange reaction
H2 + D2 = 2 HD
1. Distributions
We have 48 cards consisting of 24 red cards and 24 black cards. If we ignore the
distinction between spades, clubs, hearts, and diamonds, just calling both red cards "H”
and calling both red cards "D", we can say that there are only two distinct distributions
of cards. One possible distribution might be characterized as "H2+D2", where one hand
is both black and the other is both red. The other distinct distribution is where there are
two hands, both the same, having one black and one red card each; " 2HD" .
Over we have;
Distribution A:
H2 + D2
Distribution B:
2 HD
Trial number
Outcomes of H2
Outcomes of D2
Outcomes of 2HD
Table 1. All distributions of 48 cards.
2. Probability, P
Considering the multiplicity, "W" and probability of each outcomes, "P"
We can get the probability (Table 2) from above two formulas:
Trial number Outcomes of H2
Outcomes of D2
Outcomes of 2HD
Table 2. Probabilities of each distribution
3. The reaction quotient, Q
For a given reaction:
aA+bB=cC+dD
the reaction quotient, Q is written by:
Probability
We would apply the following formula
Trial
number
Outcomes of Outcomes of D2 Outcomes of
H2
2HD
Probability
Quotient
0.03306
0.04000
0.44444
1.0000
2.04082
4.0000
7.8400
16.0000
36.0000
100.0000
484.0000
Table 3. Quotient of the reaction H2+D2= 2 HD
4. Equilibrium Constants Described the Most Probable
Distribution
Consider again our small system contain just 48 cards, with the most probable
distribution of 6 H2, 6 D2, and 12 HD molecules.
For this equilibrium we might write
If equilibrium is really the most probable distribution, then we should be able to
substitute our values for the most probable distribution number of H2, D2, and HD
molecules for equilibrium concentrations.
If we do that, we get the equilibrium constant, K: