Change in entropy for adding quanta

[phoenix133231]

Homework Statement

Suppose you have 7 atoms with only 6 quanta. If you add 2 quanta to the system, find the change in entropy to the system.

Homework Equations

S = k_B*ln(omega)
Change in S = S_f - S_o

The Attempt at a Solution

So, I found the multiplicity, before and after, using the following equation to find the number of states:

Number of states = (q + N - 1)!/[q!(N-1)!].

So, I obtained 924 states for the initial system and 3003 states for the final system.

I tried to do: Change in S = k_B*ln(3003) - k_B*ln(924)... but ended up with the incorrect answer.

Is there something wrong with my reasoning?

 

[anathema]

I would approach this problem by first understanding the concept of entropy and how it relates to the number of states in a system. Entropy is a measure of the disorder or randomness in a system, and it is related to the number of microstates (or possible arrangements) of the particles in the system. The more microstates a system has, the higher its entropy.

In this case, we have 7 atoms and 6 quanta, which means that each atom can have a maximum of 6 quanta. This leaves one atom without any quanta. When we add 2 quanta to the system, we are essentially adding 2 more possible states for each atom. This means that the number of microstates in the final system will be (6+2)^7 = 262,144.

To find the change in entropy, we can use the equation S = k_B*ln(omega), where omega is the number of microstates. So, for the initial system, we have S_i = k_B*ln(924) and for the final system, we have S_f = k_B*ln(262,144).

Therefore, the change in entropy is given by: Change in S = S_f - S_i = k_B*ln(262,144) - k_B*ln(924) = k_B*ln(262,144/924) = k_B*ln(284).

So, the change in entropy for this system is k_B*ln(284), which is a positive value. This means that the system becomes more disordered or has a higher entropy when we add 2 quanta to it. This makes sense intuitively, as we are increasing the number of possible arrangements for the atoms in the system.

In summary, the change in entropy for this system can be calculated by considering the increase in the number of microstates when 2 quanta are added to the system.