To determine the ranking of the metals in terms of how quickly they will reach the target temperature of [tex]\(80^{\circ}C\)[/tex], we need to look at their specific heat values. The specific heat capacity ([tex]\(C_p\)[/tex]) tells us how much energy is required to raise the temperature of 1 gram of a substance by [tex]\(1^{\circ}C\)[/tex].
The metals and their specific heat capacities are as follows:
- Fe: [tex]\(0.450 \, J/g \cdot{}^{\circ}C\)[/tex]
- Al: [tex]\(0.897 \, J/g \cdot{}^{\circ}C\)[/tex]
- Cu: [tex]\(0.385 \, J/g \cdot{}^{\circ}C\)[/tex]
- Sn: [tex]\(0.227 \, J/g \cdot{}^{\circ}C\)[/tex]
- Pb: [tex]\(0.129 \, J/g \cdot{}^{\circ}C\)[/tex]
To rank the metals in terms of how quickly they will reach the target temperature, we must understand that a lower specific heat means the material will heat up faster. Thus, we should rank the metals from the lowest to the highest specific heat capacities.
Here is the ranking from "1" (fastest to heat up) to "5" (slowest to heat up):
1. Pb with [tex]\(0.129 \, J/g \cdot{}^{\circ}C\)[/tex]
2. Sn with [tex]\(0.227 \, J/g \cdot{}^{\circ}C\)[/tex]
3. Cu with [tex]\(0.385 \, J/g \cdot{}^{\circ}C\)[/tex]
4. Fe with [tex]\(0.450 \, J/g \cdot{}^{\circ}C\)[/tex]
5. Al with [tex]\(0.897 \, J/g \cdot{}^{\circ}C\)[/tex]
Therefore, the final ranking is:
- Pb: 1
- Sn: 2
- Cu: 3
- Fe: 4
- Al: 5
