To determine which substance has the smallest heat capacity, let's carefully compare the specific heat capacities ([tex]\( C_p \)[/tex]) provided for each substance. Here are the values:
[tex]\[
\begin{array}{|r|c|}
\hline
\text{Substance} & C_p (J/g \cdot {}^\circ C) \\
\hline
\text{H}_2\text{O (l)} & 4.18 \\
\text{H}_2\text{O (s)} & 2.03 \\
\text{H}_2\text{O (g)} & 2.08 \\
\text{Fe (s)} & 0.450 \\
\text{Al (s)} & 0.897 \\
\text{Cu (s)} & 0.385 \\
\text{Sn (s)} & 0.227 \\
\text{Pb (s)} & 0.129 \\
\text{Au (s)} & 0.129 \\
\text{Hg (l)} & 0.140 \\
\hline
\end{array}
\][/tex]
To find the smallest heat capacity:
1. Compare each value to determine the smallest one.
Start comparing:
- 4.18 (H[tex]\(_2\)[/tex]O (l)) is much larger.
- 2.03 (H[tex]\(_2\)[/tex]O (s)) is still high.
- 2.08 (H[tex]\(_2\)[/tex]O (g)) is also high.
- 0.450 (Fe (s)) is smaller, but there may be smaller values.
- 0.897 (Al (s)) is larger than 0.450.
- 0.385 (Cu (s)) is smaller but has lower values remaining.
- 0.227 (Sn (s)) is smaller still.
- 0.129 (Pb (s)) is the smallest so far.
- 0.129 (Au (s)) is equal to 0.129 (Pb (s)).
- 0.140 (Hg (l)) is slightly larger than 0.129.
After comparing all values, the two smallest heat capacities are for Pb (s) and Au (s), both having a specific heat capacity of 0.129 [tex]\(J/g \cdot {}^\circ C\)[/tex].
In conclusion, both [tex]\(\textbf{Pb (s)}\)[/tex] (lead) and [tex]\(\textbf{Au (s)}\)[/tex] (gold) have the smallest heat capacity among the listed substances, with a specific heat value of [tex]\(0.129 \, \text{J/g} \cdot {}^\circ C\)[/tex]. However, typically in such questions, if more than one substance shares the smallest value, it's conventional to list the first one in the provided list hence the answer is:
[tex]\[
\boxed{\text{Pb (s)}}
\][/tex]
