= Solution
\b[Sample A is the Cu–Au alloy.] At the indicated copper-rich composition, the supplied Cu–Au diagram permits one copper-rich solid-solution phase. The Cu–Ag diagram instead gives two equilibrium solid phases at room <temperature>. A single consistent set of FCC peaks therefore identifies A under the stated equilibrium assumptions; similar pure-metal lattice parameters alone would not distinguish <gold> from silver.
Use <Vegard's law> as an approximate atomic-fraction interpolation:
$$
a\simeq(1-x_{\rm Au})a_{\rm Cu}+x_{\rm Au}a_{\rm Au}.
$$
With the higher-angle estimate from part(a),
$$
x_{\rm Au}=\frac{0.363857-0.3615}{0.4078-0.3615}\simeq0.0509.
$$
This is about $5.1$ atomic percent, not weight percent. Using rounded molar masses $M_{\rm Au}\simeq197$ and $M_{\rm Cu}\simeq63.55\,\mathrm{g\,mol^{-1}}$,
$$
\boxed{w_{\rm Au}=\frac{x_{\rm Au}M_{\rm Au}}
{x_{\rm Au}M_{\rm Au}+(1-x_{\rm Au})M_{\rm Cu}}
\simeq0.143,\quad\text{about }14\,\mathrm{wt\%\ Au}.}
$$
Using the first peak alone instead gives roughly $12.6\,\mathrm{wt\%}$. The justified conclusion is therefore about \b[$13$–$14\,\mathrm{wt\%}$ <gold>], with uncertainty from angle rounding, calibration and the approximate linear lattice-composition relation.
Back to article page
