To determine which substance shows the smallest temperature change upon gaining 200.0 J of heat, we need to calculate the temperature change ([tex]\(\Delta T\)[/tex]) for each substance using the formula:
[tex]\[
\Delta T = \frac{Q}{m \cdot C}
\][/tex]
where:
- [tex]\(Q\)[/tex] is the heat added (200.0 J)
- [tex]\(m\)[/tex] is the mass of the substance in grams
- [tex]\(C\)[/tex] is the specific heat capacity in [tex]\(J/(g \cdot °C)\)[/tex]
Let's analyze the given data to find [tex]\(\Delta T\)[/tex] for each substance:
1. Water:
- Mass: [tex]\(50.0 \, g\)[/tex]
- Specific heat capacity ([tex]\(C_{\text{water}}\)[/tex]): [tex]\(4.18 \, J/(g \cdot °C)\)[/tex]
- [tex]\(\Delta T_{\text{water}} = \frac{200.0}{50.0 \cdot 4.18} \approx 0.957°C\)[/tex]
2. Iron (Fe):
- Mass: [tex]\(50.0 \, g\)[/tex]
- Specific heat capacity ([tex]\(C_{\text{Fe}}\)[/tex]): [tex]\(0.449 \, J/(g \cdot °C)\)[/tex]
- [tex]\(\Delta T_{\text{Fe}} = \frac{200.0}{50.0 \cdot 0.449} \approx 8.909°C\)[/tex]
3. Granite:
- Mass: [tex]\(50.0 \, g\)[/tex]
- Specific heat capacity ([tex]\(C_{\text{granite}}\)[/tex]): [tex]\(0.79 \, J/(g \cdot °C)\)[/tex]
- [tex]\(\Delta T_{\text{granite}} = \frac{200.0}{50.0 \cdot 0.79} \approx 5.063°C\)[/tex]
4. Silver (Ag):
- Mass: [tex]\(50.0 \, g\)[/tex]
- Specific heat capacity ([tex]\(C_{\text{Ag}}\)[/tex]): [tex]\(0.235 \, J/(g \cdot °C)\)[/tex]
- [tex]\(\Delta T_{\text{Ag}} = \frac{200.0}{50.0 \cdot 0.235} \approx 17.021°C\)[/tex]
5. Lead (Pb):
- Mass: [tex]\(25.0 \, g\)[/tex]
- Specific heat capacity ([tex]\(C_{\text{Pb}}\)[/tex]): [tex]\(0.128 \, J/(g \cdot °C)\)[/tex]
- [tex]\(\Delta T_{\text{Pb}} = \frac{200.0}{25.0 \cdot 0.128} \approx 62.5°C\)[/tex]
After calculating these temperature changes, we find:
[tex]\[
\Delta T_{\text{water}} \approx 0.957°C
\][/tex]
[tex]\[
\Delta T_{\text{Fe}} \approx 8.909°C
\][/tex]
[tex]\[
\Delta T_{\text{granite}} \approx 5.063°C
\][/tex]
[tex]\[
\Delta T_{\text{Ag}} \approx 17.021°C
\][/tex]
[tex]\[
\Delta T_{\text{Pb}} \approx 62.5°C
\][/tex]
Comparing all these values, we can see that the smallest temperature change occurs in water, with [tex]\(\Delta T_{\text{water}} \approx 0.957°C\)[/tex].
Therefore, the substance that shows the smallest temperature change upon gaining 200.0 J of heat is 50.0 g of water.
