Naturally occurring lead has the following isotopic abundances: $$ \begin{array}{lcc} \hline \text { Isotope } & \text { Abundance (\%) } & \text { Atomic mass (u) } \\ \hline{ }^{204} \mathrm{~Pb} & 1.4 & 203.9730 \\ { }^{206} \mathrm{~Pb} & 24.1 & 205.9744 \\ { }^{207} \mathrm{~Pb} & 22.1 & 206.9759 \\ { }^{208} \mathrm{~Pb} & 52.4 & 207.9766 \\ \hline \end{array} $$ (a) What is the average atomic mass of \(\mathrm{Pb}\) ? (b) Sketch the mass spectrum of \(\mathrm{Pb}\).

To find the average atomic mass of lead, we need to multiply each isotope's abundance by its atomic mass and then sum them up. This is the weighted average atomic mass. Let's denote the isotopes as follows: - Isotope 1: Pb-204 with abundance 1.4% and atomic mass 203.9730 u - Isotope 2: Pb-206 with abundance 24.1% and atomic mass 205.9744 u - Isotope 3: Pb-207 with abundance 22.1% and atomic mass 206.9759 u - Isotope 4: Pb-208 with abundance 52.4% and atomic mass 207.9766 u Now we can calculate the average atomic mass: Average Atomic Mass \(= (1.4\% \cdot 203.9730\,u) + (24.1\% \cdot 205.9744\,u) + (22.1\% \cdot 206.9759\,u) + (52.4\% \cdot 207.9766\,u)\) Before we proceed, we need to convert the percentages to their decimal forms: 1.4% = 0.014 24.1% = 0.241 22.1% = 0.221 52.4% = 0.524 Now we plug in the decimal values in the formula: Average Atomic Mass \(= (0.014 \cdot 203.9730) + (0.241\cdot 205.9744) + (0.221\cdot 206.9759) + (0.524\cdot 207.9766)\) Solve for the average atomic mass: Average Atomic Mass \(= 2.8556 + 49.6396 + 45.7404 + 109.0006 = 207.2362\, u\) (a) The average atomic mass of lead (Pb) is 207.2362 u.

To sketch the mass spectrum of lead, we need to plot the isotopes on the x-axis and their relative abundances on the y-axis. Each isotope will be represented by a vertical bar with a height proportional to its abundance. Here is how the mass spectrum should look like: - The x-axis represents the atomic mass, ranging from approximately 203.5 u to 208.5 u, with labels on the isotope peaks. - The y-axis represents the relative abundance, ranging from 0 to 60 %. - Four vertical bars mark the isotopes with their respective atomic masses (203.9730 u, 205.9744 u, 206.9759 u, and 207.9766 u) and abundances (1.4 %, 24.1 %, 22.1 %, 52.4 %), with the highest bar being the most abundant isotope (Pb-208). (b) The mass spectrum of lead (Pb) will show four peaks corresponding to its four isotopes, with the highest peak representing Pb-208. The heights of the peaks correspond to their relative abundances.