A tabulation of data lists the following equation for calculating the densities \((d)\) of solutions of naphthalene in benzene at \(30^{\circ} \mathrm{C}\) as a function of the mass percent of naphthalene. $$d\left(\mathrm{g} / \mathrm{cm}^{3}\right)=\frac{1}{1.153-1.82 \times 10^{-3}(\% \mathrm{N})+1.08 \times 10^{-6}(\% \mathrm{N})^{2}}$$ Use the equation above to calculate (a) the density of pure benzene at \(30^{\circ} \mathrm{C} ;\) (b) the density of pure naphthalene at \(30^{\circ} \mathrm{C} ;\) (c) the density of solution at \(30^{\circ} \mathrm{C}\) that is 1.15\% naphthalene; (d) the mass percent of naphthalene in a solution that has a density of \(0.952 \mathrm{g} / \mathrm{cm}^{3}\) at \(30^{\circ} \mathrm{C} .[\text { Hint: For }(\mathrm{d}),\) you need to use the quadratic formula. See Section A-3 of Appendix A.]

Step by step solution

01

Calculating the density of pure benzene

For pure benzene, there is no naphthalene involved, hence, the mass percent of naphthalene (\(% N\)) is 0. Substituting \%N = 0 into the density equation \(d=(1)/(1.153-1.82 \times 10^{-3}(\%N)+1.08 \times 10^{-6}(\%N)^{2})\), gives the density of pure benzene.

02

Calculating the density of pure naphthalene

For pure naphthalene, the mass percent of naphthalene (\(% N\)) is 100. Substituting \%N = 100 into the density equation \(d=(1)/(1.153-1.82 \times 10^{-3}(\%N)+1.08 \times 10^{-6}(\%N)^{2})\), gives the density of pure naphthalene.

03

Calculating the density of the solution with 1.15% naphthalene

Here, the mass percent of naphthalene (\(% N\)) is 1.15. Substituting \%N = 1.15 into the density equation \(d=(1)/(1.153-1.82 \times 10^{-3}(\%N)+1.08 \times 10^{-6}(\%N)^{2})\), gives the density of the solution.

04

Finding the mass percent of naphthalene in a solution with a given density

Here, the density of the solution (\(d\)) is given and the mass percent of naphthalene (\(% N\)) needs to be determined. Thus, the equation needs to be rewritten as \(%N = (1.153d -1)/(1.82 \times 10^{-3} + 1.08 \times 10^{-6}d)\) and solved using the quadratic formula to find \%N.

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