Questions: Entropy and Free Energy Calculating absolute entropy using the Boltzmann hypothesis An oxygen (O2) molecule is adsorbed onto a small patch of the surface of a catalyst. It's known that the molecule is adsorbed on 1 of 64 possible sites for adsorption. Calculate the entropy of this system. Round your answer to 3 significant digits, and be sure it has the correct unit symbol. An O2 molecule adsorbed at one site on a surface.

Solution Steps

The Boltzmann entropy formula is given by:

\[ S = k \ln W \]

where \( S \) is the entropy, \( k \) is the Boltzmann constant (\(1.3807 \times 10^{-23} \, \text{J/K}\)), and \( W \) is the number of microstates.

In this problem, the oxygen molecule can be adsorbed on 1 of 64 possible sites. Therefore, the number of microstates \( W \) is 64.

Substitute the values into the Boltzmann entropy formula:

\[ S = k \ln W = (1.3807 \times 10^{-23} \, \text{J/K}) \ln(64) \]

Calculate \( \ln(64) \):

\[ \ln(64) = \ln(2^6) = 6 \ln(2) \approx 6 \times 0.6931 = 4.1586 \]

Now, calculate the entropy:

\[ S = (1.3807 \times 10^{-23} \, \text{J/K}) \times 4.1586 \approx 5.742 \times 10^{-23} \, \text{J/K} \]

Final Answer