Consider a system of 3 molecules, A, B, and C. Suppose the 3 units of energy can be distributed over the three atoms where each atom can have no energy, one unit, two units or 3 units of energy. Assume each of the 10 possible arrangements is equally probable. What's probability that molecule C has some energy?
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To find the probability that molecule C has some energy, we first need to determine the total number of possible arrangements.
In this system, each atom can have 0, 1, 2, or 3 units of energy. So, using the multiplication principle, we multiply the number of possibilities for each atom to find the total arrangements.
For molecule A, there are 4 possibilities (0, 1, 2, or 3 units of energy).
For molecule B, there are also 4 possibilities.
For molecule C, since we specifically want it to have some energy, it can have 1, 2, or 3 units of energy. So there are 3 possibilities.
Using the multiplication principle, the total number of arrangements is calculated as 4 * 4 * 3 = 48.
Next, we need to determine the number of arrangements where molecule C has some energy. In this case, molecule C can have 1, 2, or 3 units of energy, while molecules A and B can have any energy configuration.
To determine the number of such arrangements, we calculate the possibilities for molecules A and B, which is again 4 * 4 = 16. Then, we multiply this number by the possibilities for molecule C, which is 3 (the number of arrangements where molecule C has some energy).
So the total number of arrangements where molecule C has some energy is 16 * 3 = 48.
Finally, we can compute the probability by dividing the number of favorable outcomes (48) by the total number of outcomes (48). Therefore, the probability that molecule C has some energy is 48/48 = 1, which means it is guaranteed that molecule C has some energy in each of the 10 possible arrangements.