### Solution
#### (a) Calculate the average atomic mass of cerium
To find the average atomic mass of cerium, we need to use the relative isotopic masses and their respective percentage abundances.
The naturally occurring isotopes and their relative isotopic masses and percentage abundances are:
[tex]\[
\begin{array}{|c|c|c|}
\hline \text{Isotope} & \text{Relative Isotopic Mass} & \text{Percentage Abundance (\%)} \\
\hline ^{136}\text{Ce} & 135.91 & 0.19 \\
\hline ^{138}\text{Ce} & 137.91 & 0.25 \\
\hline ^{140}\text{Ce} & 139.91 & 88.48 \\
\hline ^{142}\text{Ce} & 141.91 & 11.08 \\
\hline
\end{array}
\][/tex]
First, convert the percentage abundances to decimal form:
[tex]\[
\text{Decimal abundances:}
\begin{array}{l}
0.19\% \to 0.0019 \\
0.25\% \to 0.0025 \\
88.48\% \to 0.8848 \\
11.08\% \to 0.1108 \\
\end{array}
\][/tex]
Next, we calculate the average atomic mass using the formula:
[tex]\[
\text{Average Atomic Mass} = \sum (\text{Relative Isotopic Mass} \times \text{Decimal Abundance})
\][/tex]
Substituting the values:
[tex]\[
\begin{align*}
\text{Average Atomic Mass} &= (135.91 \times 0.0019) + (137.91 \times 0.0025) + (139.91 \times 0.8848) + (141.91 \times 0.1108) \\
&= 0.258229 + 0.344775 + 123.853208 + 15.662788 \\
&= 140.119
\end{align*}
\][/tex]
Therefore, the average atomic mass of cerium is 140.119.
#### (b) Calculate the relative atomic mass of cerium
The relative atomic mass of cerium, which is its average atomic mass calculated above, is 140.119.
#### (c) Calculate the mole fraction of methanol in the solution
Given:
- Mass of methanol ([tex]\( CH_3OH \)[/tex]) = 32.0 g
- Mass of water ([tex]\( H_2O \)[/tex]) = 72.0 g
- Molar mass of methanol = 32.04 g/mol
- Molar mass of water = 18.015 g/mol
First, calculate the number of moles of methanol and water:
[tex]\[
\text{Moles of methanol} = \frac{\text{Mass}}{\text{Molar Mass}} = \frac{32.0 \text{ g}}{32.04 \text{ g/mol}} = 0.9987515605493134 \text{ mol}
\][/tex]
[tex]\[
\text{Moles of water} = \frac{\text{Mass}}{\text{Molar Mass}} = \frac{72.0 \text{ g}}{18.015 \text{ g/mol}} = 3.996669442131557 \text{ mol}
\][/tex]
Next, calculate the total number of moles in the solution:
[tex]\[
\text{Total moles} = \text{Moles of methanol} + \text{Moles of water} = 0.9987515605493134 + 3.996669442131557 = 4.9954210026808704 \text{ mol}
\][/tex]
Finally, calculate the mole fraction of methanol:
[tex]\[
\text{Mole fraction of methanol} = \frac{\text{Moles of methanol}}{\text{Total moles}} = \frac{0.9987515605493134}{4.9954210026808704} = 0.19993341102047613
\][/tex]
Therefore, the mole fraction of methanol in the solution is 0.19993341102047613.
### Summary
- (a) The average atomic mass of cerium is 140.119.
- (b) The relative atomic mass of cerium is 140.119.
- (c) The mole fraction of methanol in the solution is 0.19993341102047613.
