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## Q. 3.17

Lycopene (molar mass 536.88 g/mol) and cymene (134.22 g/mol) are natural products found in tomatoes and cumin (a common spice), respectively (Figure 3.30). Lycopene in the diet has been shown to reduce the risk of prostate cancer, whereas cumin is a home remedy for stomach ailments. Both compounds are 89.49% C and 10.51% H by mass. What are the empirical and molecular formulas of lycopene and cymene?

## Verified Solution

Collect and Organize The problem gives the percent composition and molar masses of two compounds. We want to determine both the empirical and molecular formulas of lycopene and cymene.

Analyze The data on percent composition lead directly to the empirical formula by using the procedure described in Section 3.6. The molar masses of lycopene and cymene in g/mol are numerically the same as their molecular masses in amu and can be used to determine the value of the multipliers needed to convert their empirical formulas into molecular formulas.

Solve Assuming a 100 g sample, we have 89.49 g of C and 10.51 g of H, which we convert to moles:

$89.49 \sout{g C} \times \frac{1 mol C}{12.01 \sout{g C}}=7.451 mol C$

$10.51 \sout{g H} \times \frac{1 mol H}{1.008 \sout{g H}}=10.43 mol H$

The mole ratio of C to H is 7.451:10.43, or 1:1.4. We can convert the fractional mole ratio to a ratio of whole numbers by multiplying by 5:

$C:H= 5 × (1:1.4) = 5:7$

which means the empirical formula is $C_{5}H_{7}$.
To determine the molecular formula, we first determine the molar mass of the empirical formula $C_{5}H_{7}$:

$5 \sout{mol}(12.01 g/\sout{mol}) + 7 \sout{mol}(1.008 g/\sout{mol}) = 67.11 g$

Next we divide the molar masses for lycopene and cymene by their respective empirical formula masses to determine the values of the multiplier n:

$n=\frac{molar mass lycopene}{empirical formula mass}=\frac{536.88 \sout{g/mol}}{67.11 \sout{g/mol}}=8$

$n=\frac{molar mass cymene}{empirical formula mass}=\frac{134.22 \sout{g/mol}}{67.11 \sout{g/mol}}=2$

The molecular formula of lycopene is

$(C_{5}H_{7})_{n}=(C_{5}H_{7})_{8}=C_{40}H_{56}$

and the molecular formula of cymene is

$(C_{5}H_{7})_{n}=(C_{5}H_{7})_{2}=C_{10}H_{14}$

Think About It The percent composition of a compound reveals only the empirical formula.
All compounds with the same empirical formula must have the same percent composition.