To determine the final temperature of the water sample after it absorbs 13.98 kJ of heat, we'll proceed with the following steps:
1. Convert the heat energy to joules:
- Given that 1 kJ = 1000 J, we have:
[tex]\[
\text{heat energy} = 13.98 \text{kJ} \times 1000 = 13980 \text{J}
\][/tex]
2. Determine the specific heat capacity of water:
- The specific heat capacity ([tex]\( c \)[/tex]) of water is a known constant:
[tex]\[
c = 4.18 \text{J/g°C}
\][/tex]
3. Find the mass of the water sample:
- The mass ([tex]\( m \)[/tex]) of the water sample is given as:
[tex]\[
m = 705.2 \text{g}
\][/tex]
4. Calculate the change in temperature ([tex]\( \Delta T \)[/tex]):
- The formula to calculate the change in temperature when a substance absorbs heat is given by:
[tex]\[
Q = m \cdot c \cdot \Delta T
\][/tex]
where [tex]\( Q \)[/tex] is the heat absorbed. Rearranging this formula to solve for [tex]\( \Delta T \)[/tex]:
[tex]\[
\Delta T = \frac{Q}{m \cdot c}
\][/tex]
- Plug in the values:
[tex]\[
\Delta T = \frac{13980 \text{J}}{705.2 \text{g} \times 4.18 \text{J/g°C}}
\][/tex]
- Perform the division to find [tex]\( \Delta T \)[/tex]:
[tex]\[
\Delta T = 4.742622812897763 \text{°C}
\][/tex]
5. Calculate the final temperature:
- The initial temperature ([tex]\( T_{\text{initial}} \)[/tex]) of the water is given as:
[tex]\[
T_{\text{initial}} = 7 \text{°C}
\][/tex]
- To find the final temperature ([tex]\( T_{\text{final}} \)[/tex]), add the change in temperature ([tex]\( \Delta T \)[/tex]) to the initial temperature:
[tex]\[
T_{\text{final}} = T_{\text{initial}} + \Delta T
\][/tex]
- Substituting the values:
[tex]\[
T_{\text{final}} = 7 \text{°C} + 4.742622812897763 \text{°C}
\][/tex]
[tex]\[
T_{\text{final}} = 11.742622812897764 \text{°C}
\][/tex]
Hence, the final temperature of the water sample after absorbing 13.98 kJ of heat is approximately 11.74°C.
