Chapter 2, Problem 2.1CYU

Check Your Understanding

The recommended daily amount of vitamin B₃ or niacin, C6NH₅ O₂, for women who are not pregnant or nursing, is 14 mg. Find the number of molecule of niacin in that amount.

To determine

The number of molecules of niacin in $14\text{mg}$ of ${\text{C}}_{\text{6}}{\text{NH}}_{\text{5}}{\text{O}}_{\text{2}}$.

Answer to Problem 2.1CYU

The number of molecules of niacin in $14\text{mg}$ of ${\text{C}}_{\text{6}}{\text{NH}}_{\text{5}}{\text{O}}_{\text{2}}$ is $6.85\times {10}^{19}\text{molecules}$.

Explanation of Solution

Given:

The amount of vitamin ${\text{B}}_{\text{3}}$, niacin or ${\text{C}}_{\text{6}}{\text{NH}}_{\text{5}}{\text{O}}_{\text{2}}$ is $m=14\text{mg}$.

The number of mole of carbon is $n_c=6$.

The number of mole of hydrogen is $n_h=5$.

The number of mole of nitrogen is $n_{nit}=1\text{mole}$.

The number of mole of oxygen is $n_o=2\text{mole}$.

Formula used:

The molar mass of niacin is given by

  $m_n=n_c{m}_c+n_h{m}_h+n_{nit}{m}_{nit}+n_o{m}_o$

The number of moles is given by

  $n=\frac{m}{m_m}$

The number of molecule is given by

  $N=n{N}_A$

Calculation:

The molar mass of carbon is $m_c=12\text{g}/\text{mole}$.

The molar mass of hydrogen is $m_h=1\text{g}/\text{mole}$.

The molar mass of nitrogen is $m_{nit}=14\text{g}/\text{mole}$.

The molar mass of oxygen is $m_o=16\text{g}/\text{mole}$.

The molar mass of niacin is calculated as follows:

  $\begin{array}{c}{m}_n=n_c{m}_c+n_h{m}_h+n_{nit}{m}_{nit}+n_o{m}_o\\ =6\left(12\text{g}/\text{mole}\right)+5\left(1\text{g}/\text{mole}\right)+1\left(14\text{g}/\text{mole}\right)+2\left(16\text{g}/\text{mole}\right)\\ =\left(72\text{g}/\text{mole}\right)+\left(5\text{g}/\text{mole}\right)+\left(14\text{g}/\text{mole}\right)+\left(32\text{g}/\text{mole}\right)\\ =123\text{g}/\text{mole}\end{array}$

The number of moles is calculated as follows:

  $\begin{array}{c}n=\frac{m}{m_m}\\ =\frac{\left(14\text{mg}\right)}{\left(123\text{g}/\text{mole}\right)}\\ =\frac{\left(14\text{mg}\right)\left(\frac{1\text{g}}{1000\text{mg}}\right)}{\left(123\text{g}/\text{mole}\right)}\\ =1.138\times {10}^{-4}\text{mole}\end{array}$

The Avogadro number is $N_A=6.02\times {10}^{23}\text{molecules}/\text{mole}$.

The number of molecule is calculated as follows:

  $\begin{array}{c}N=n{N}_A\\ =\left(1.138\times {10}^{-4}\text{mole}\right)\left(6.02\times {10}^{23}\text{molecules}/\text{mole}\right)\\ =6.85\times {10}^{19}\text{molecules}\end{array}$

Conclusion:

Thus, the number of molecules of niacin in $14\text{mg}$ of ${\text{C}}_{\text{6}}{\text{NH}}_{\text{5}}{\text{O}}_{\text{2}}$ is $6.85\times {10}^{19}\text{molecules}$.