\(20\) g of gold (Au) and \(20\) g of silver (Ag) are mixed to form a single phase solid solution (assume ideal mixing). The atomic weight of Au is \(197\) g/mol and the atomic weight of Ag is \(108\) g/mol. The universal gas constant \(R = 8.314\ \text{J/mol-K}\). Find the total entropy of mixing (rounded off to two decimal places), in J/K.
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Correct Answer: 1.5
Solution and Explanation
The number of moles is mass divided by atomic weight.
For gold:
\[ n_{Au} = \frac{20}{197} = 0.10152\ \text{mol} \]
For silver:
\[ n_{Ag} = \frac{20}{108} = 0.18519\ \text{mol} \]
Step 2: Find the total number of moles.
\[ n_{total} = n_{Au} + n_{Ag} = 0.10152 + 0.18519 = 0.28671\ \text{mol} \]
Step 3: Find the mole fractions.
\[ x_{Au} = \frac{n_{Au}}{n_{total}} = \frac{0.10152}{0.28671} = 0.3541 \]
\[ x_{Ag} = \frac{n_{Ag}}{n_{total}} = \frac{0.18519}{0.28671} = 0.6459 \]
Check: \(0.3541 + 0.6459 = 1.0000\), as expected.
Step 4: Recall the entropy of mixing for an ideal solid solution.
For an ideal solution, the molar entropy of mixing is
\[ \Delta S_{mix} = -R\left(x_{Au}\ln x_{Au} + x_{Ag}\ln x_{Ag}\right) \]
This gives the entropy gained per mole of the mixture from randomly arranging the two kinds of atoms on the lattice sites.
Step 5: Compute the logarithm terms.
\[ \ln x_{Au} = \ln(0.3541) = -1.0382 \]
\[ \ln x_{Ag} = \ln(0.6459) = -0.4371 \]
So,
\[ x_{Au}\ln x_{Au} = 0.3541 \times (-1.0382) = -0.3676 \]
\[ x_{Ag}\ln x_{Ag} = 0.6459 \times (-0.4371) = -0.2823 \]
Adding these,
\[ x_{Au}\ln x_{Au} + x_{Ag}\ln x_{Ag} = -0.3676 - 0.2823 = -0.6499 \]
Step 6: Find the molar entropy of mixing.
\[ \Delta S_{mix} = -8.314 \times (-0.6499) = 5.4037\ \text{J/mol-K} \]
Step 7: Scale up by the total number of moles.
The value above is entropy per mole of the mixture, so multiply by the total moles present to get the total entropy of mixing.
\[ \Delta S_{mix,total} = 5.4037 \times 0.28671 = 1.5493\ \text{J/K} \]
Final Answer:
Rounded to two decimal places, the total entropy of mixing is 1.55 J/K.
\[ \boxed{\Delta S_{mix,total} \approx 1.55\ \text{J/K}} \]
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