Question

For a certain metal, the stiffness of the interatomic bond and the mass of one atom are such that the spacing of the quantum oscillator energy levels is 8.0e-23 J. A nanoparticle of this metal consisting of 8 atoms has a total thermal energy of 112e-23 J. What is the entropy of this nanoparticle?

Answer 1

The entropy of a nanoparticle can be calculated based on the number of atoms and the spacing of the quantum oscillator energy levels. In this case, the nanoparticle consists of 8 atoms and has a total thermal energy of 112e-23 J.

Step-by-step explanation:

1. Determine the number of energy levels accessible:

Total thermal energy (E_total) = 112e-23 J

Energy level spacing (dE) = 8.0e-23 J

Number of accessible energy levels (n_levels) = E_total / dE + 1 (add 1 to account for the ground state)

n_levels = 112e-23 J / 8.0e-23 J + 1 = 15

2. Calculate the entropy using Boltzmann's formula:

Boltzmann constant (k_B) = 1.38064852e-23 J/K

Number of atoms (N) = 8

Entropy (S) = k_B × N × log2(n_levels)

S = 1.38064852e-23 J/K × 8 × log2(15)

S ≈ 4.32e-22 J/K

Therefore, the entropy of the nanoparticle is approximately 4.32e-22 J/K.