I think this should be easy but im stuck. Please help:
Suppose a compound could point in any of two directions in the solid and still have the same energy. How many molecules would there be if the total entropy of a solid sample of this compound was 8.174e-9J/K
I used S= klnW and got lnW = 5.92e14. How do i get the number of molecules from here?
W = 2ⁿ , where n = number of molecules
lnW = n • ln2 = 5.921 × 10¹⁴ ⇒
n = 8.542 × 10¹⁴ molecules
To find the number of molecules, you'll need to use the relation between entropy, number of microstates, and Boltzmann's constant. Here's how you can proceed:
1. Start with the equation you already have: lnW = 5.92e14.
2. Rearrange the equation to isolate W: W = e^(lnW).
3. Substitute the value of lnW into the equation: W = e^(5.92e14).
4. Now, recall that the total entropy S is given by S = k lnW, where k is Boltzmann's constant (approximately 1.38e-23 J/K).
5. Rearrange the entropy equation to isolate W: W = e^(S / k).
6. Substitute the given entropy value into the equation: W = e^(8.174e-9 J/K / (1.38e-23 J/K)).
7. Calculate the value of W using a calculator or software, if needed.
8. Finally, the value of W represents the number of microstates of the system. Since each microstate corresponds to a distinct arrangement of the molecules, the number of molecules (N) should be equal to W.
Therefore, the number of molecules in the solid sample of the compound would be equal to the value of W obtained from step 7.