To determine the amount of heat gained by the gold, we can use the formula for heat transfer in the context of specific heat capacity. The formula is:
[tex]\[ Q = mc\Delta T \][/tex]
Where:
- [tex]\( Q \)[/tex] is the amount of heat gained (in joules)
- [tex]\( m \)[/tex] is the mass of the gold (in grams)
- [tex]\( c \)[/tex] is the specific heat capacity (in J/(g·°C))
- [tex]\( \Delta T \)[/tex] is the change in temperature (in °C)
Let's calculate it step-by-step:
1. Determine the change in temperature ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 47\,^\circ\text{C} - 35\,^\circ\text{C} \][/tex]
[tex]\[ \Delta T = 12\,^\circ\text{C} \][/tex]
2. Substitute the known values into the formula:
[tex]\[ m = 10.0 \text{ g} \][/tex]
[tex]\[ c = 0.03 \text{ J/(g·°C)} \][/tex]
[tex]\[ \Delta T = 12\,^\circ\text{C} \][/tex]
Now, plug these values into the heat transfer formula:
[tex]\[ Q = (10.0 \text{ g}) \times (0.03 \text{ J/(g·°C)}) \times (12\,^\circ\text{C}) \][/tex]
3. Perform the multiplication:
[tex]\[ Q = 10.0 \times 0.03 \times 12 \][/tex]
[tex]\[ Q = 10.0 \times 0.36 \][/tex]
[tex]\[ Q = 3.6 \text{ J} \][/tex]
So, the amount of heat gained by the gold is:
[tex]\[ \boxed{3.6 \text{ J}} \][/tex]
