To find out how much energy as heat a mercury sample gains when heated from 5.0°C to 55.0°C, we need to use the concept of molar heat capacity. Here is a detailed step-by-step solution:
### Given Data
- Molar heat capacity ([tex]\(C_m\)[/tex]) of mercury: [tex]\(27.8 \, \text{J/(mol·K)}\)[/tex]
- Mass ([tex]\(m\)[/tex]) of the mercury sample: [tex]\(10.15 \, \text{g}\)[/tex]
- Initial temperature ([tex]\(T_i\)[/tex]): [tex]\(5.0 \, \text{°C}\)[/tex]
- Final temperature ([tex]\(T_f\)[/tex]): [tex]\(55.0 \, \text{°C}\)[/tex]
- Molar mass of mercury ([tex]\(M\)[/tex]): [tex]\(200.59 \, \text{g/mol}\)[/tex]
### Steps to Find the Heat Energy Gain
1. Convert Mass to Moles:
We need to convert the mass of the mercury sample to moles using the molar mass.
[tex]\[
\text{Number of moles (n)} = \frac{\text{mass (m)}}{\text{molar mass (M)}}
\][/tex]
[tex]\[
n = \frac{10.15 \, \text{g}}{200.59 \, \text{g/mol}}
\][/tex]
[tex]\[
n \approx 0.0506 \, \text{mol}
\][/tex]
2. Calculate the Temperature Change:
We need to find the change in temperature ([tex]\(\Delta T\)[/tex]).
[tex]\[
\Delta T = T_f - T_i
\][/tex]
[tex]\[
\Delta T = 55.0 \, \text{°C} - 5.0 \, \text{°C}
\][/tex]
[tex]\[
\Delta T = 50.0 \, \text{°C}
\][/tex]
3. Calculate the Heat Energy Gain:
Using the formula for heat energy ([tex]\(Q\)[/tex]), which is given by:
[tex]\[
Q = n \cdot C_m \cdot \Delta T
\][/tex]
Substituting the values we have:
[tex]\[
Q = 0.0506 \, \text{mol} \cdot 27.8 \, \text{J/(mol·K)} \cdot 50.0 \, \text{K}
\][/tex]
4. Perform the Multiplication:
[tex]\[
Q = 0.0506 \cdot 27.8 \cdot 50.0
\][/tex]
[tex]\[
Q \approx 70.4 \cdot 50.0
\][/tex]
[tex]\[
Q \approx 3520.0 \, \text{J}
\][/tex]
### Answer
The mercury sample gains approximately [tex]\(3520.0 \, \text{J}\)[/tex] of energy as heat.
