weird residual entropy formula from class

[Emily Vainberg 1D]

I am looking over my lecture notes and found this formula

S= n ln 2R
- can someone explain when you use this formula rather than the typical formula for residual entropy ?

typical formula:
S= Kb * ln W

[Ryan Narisma 4G]

Hello Emily Vainberg 1D! To answer your question, the equation that you wrote in your notes is specifically for 1 mol of a species that has two possible microstates. If you were to plug these conditions in the boltzmann equation (S = kB ln W) you'd get S = kB ln 2^(1 mol = 6.022 x 10^23 particles). Remember that in the boltzmann equation, degeneracy is equal to the number of microstates raised to the number of atoms, particles, etc. So if you have avogadro's number in the exponent, using laws of logarithms you can multiply that out (a log x = log x^a). The boltzmann constant times avogadro's number is the same as R. Then you'd get S = nR ln 2. The n comes into play because it depends on how many moles of the species you have. I hope this helps!