Question

{"given":"Composition of cuprous oxide sample: $\\text{Cu}_{1.92}\\text{O}_{1.00}$. The defect is a metal deficiency (non-stoichiometric) defect. Some $\\text{Cu}^+$ ions are replaced by $\\text{Cu}^{2+}$ ions to maintain electrical neutrality, resulting in fewer copper atoms than expected.","key_observation":"In ideal cuprous oxide ($\\text{Cu}_2\\text{O}$), all copper ions are $\\text{Cu}^+$. In the defective crystal, some $\\text{Cu}^+$ sites are vacant and to maintain charge neutrality, some remaining $\\text{Cu}^+$ ions are oxidised to $\\text{Cu}^{2+}$. Per formula unit with 1 oxygen ($\\text{O}^{2-}$, charge = $-2$), total positive charge must equal $+2$. We set up a charge balance equation with $x$ = moles of $\\text{Cu}^{2+}$ and $(1.92 - x)$ = moles of $\\text{Cu}^+$ to solve for the ratio.","option_analysis":[{"label":"(A)","text":"$4 : 5$","verdict":"incorrect","explanation":"From charge balance: $2x + (1.92 - x) = 2 \\Rightarrow x = 0.08$, giving $\\text{Cu}^{2+} : \\text{Cu}^+ = 0.08 : 1.84 = 1 : 23$, not $4:5$. This option is significantly off from the correct calculation."},{"label":"(B)","text":"$1 : 12.5$","verdict":"incorrect","explanation":"This ratio $1:12.5$ would imply $\\text{Cu}^{2+} = 0.08$ and $\\text{Cu}^+ = 1.00$, which does not satisfy the total copper count of 1.92 per formula unit. The charge balance calculation gives $1:23$, not $1:12.5$."},{"label":"(C)","text":"$1 : 23$","verdict":"correct","explanation":"Let $x$ = moles of $\\text{Cu}^{2+}$. Charge balance per formula unit: $2x + 1 \\times (1.92 - x) = 2 \\Rightarrow x = 0.08$. So $\\text{Cu}^+ = 1.92 - 0.08 = 1.84$. Ratio $= 0.08 : 1.84 = 1 : 23$. This is correct."},{"label":"(D)","text":"$1 : 24$","verdict":"incorrect","explanation":"A ratio of $1:24$ would require $\\text{Cu}^{2+} = 0.08$ and $\\text{Cu}^+ = 1.92$, making the total copper $2.00$, which contradicts the given formula $\\text{Cu}_{1.92}\\text{O}_{1.00}$. The correct total must be $1.92$, yielding $1:23$."}],"answer":"(C)","formula_steps":[]}