Answer :
To determine how much heat must be transferred to 3500 grams of liquid water to change its temperature from [tex]\(27^{\circ} C\)[/tex] to [tex]\(32^{\circ} C\)[/tex], we can use the formula for heat transfer:
[tex]\[ Q = m \cdot c \cdot \Delta T \][/tex]
where:
- [tex]\( Q \)[/tex] is the heat energy,
- [tex]\( m \)[/tex] is the mass of the water,
- [tex]\( c \)[/tex] is the specific heat capacity of the water,
- [tex]\( \Delta T \)[/tex] is the change in temperature.
Let's break this down step-by-step:
1. Determine the mass of the water ([tex]\(m\)[/tex]):
[tex]\[ m = 3500 \, \text{g} \][/tex]
2. Identify the specific heat capacity ([tex]\(c\)[/tex]):
[tex]\[ c = 4.186 \, \text{J/g} \cdot { }^{\circ} \text{C} \][/tex]
3. Calculate the change in temperature ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 32^{\circ}C - 27^{\circ}C \][/tex]
[tex]\[ \Delta T = 5^{\circ}C \][/tex]
4. Substitute the values into the heat transfer equation:
[tex]\[ Q = m \cdot c \cdot \Delta T \][/tex]
[tex]\[ Q = 3500 \, \text{g} \cdot 4.186 \, \text{J/g} \cdot { }^{\circ} \text{C} \cdot 5^{\circ} \text{C} \][/tex]
5. Calculate the heat transferred ([tex]\(Q\)[/tex]):
[tex]\[ Q = 3500 \cdot 4.186 \cdot 5 \][/tex]
[tex]\[ Q = 73255 \, \text{J} \][/tex]
Hence, the amount of heat that must be transferred to 3500 grams of liquid water to change its temperature from [tex]\(27^{\circ}C\)[/tex] to [tex]\(32^{\circ}C\)[/tex] is [tex]\(73255 \, \text{J}\)[/tex].
Comparing this value to the given options:
A. [tex]\(2900 \, \text{J}\)[/tex]
B. [tex]\(73,000 \, \text{J}\)[/tex]
C. [tex]\(170 \, \text{J}\)[/tex]
D. [tex]\(470,000 \, \text{J}\)[/tex]
The closest and most correct option is:
B. [tex]\(73,000 \, \text{J}\)[/tex]
[tex]\[ Q = m \cdot c \cdot \Delta T \][/tex]
where:
- [tex]\( Q \)[/tex] is the heat energy,
- [tex]\( m \)[/tex] is the mass of the water,
- [tex]\( c \)[/tex] is the specific heat capacity of the water,
- [tex]\( \Delta T \)[/tex] is the change in temperature.
Let's break this down step-by-step:
1. Determine the mass of the water ([tex]\(m\)[/tex]):
[tex]\[ m = 3500 \, \text{g} \][/tex]
2. Identify the specific heat capacity ([tex]\(c\)[/tex]):
[tex]\[ c = 4.186 \, \text{J/g} \cdot { }^{\circ} \text{C} \][/tex]
3. Calculate the change in temperature ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 32^{\circ}C - 27^{\circ}C \][/tex]
[tex]\[ \Delta T = 5^{\circ}C \][/tex]
4. Substitute the values into the heat transfer equation:
[tex]\[ Q = m \cdot c \cdot \Delta T \][/tex]
[tex]\[ Q = 3500 \, \text{g} \cdot 4.186 \, \text{J/g} \cdot { }^{\circ} \text{C} \cdot 5^{\circ} \text{C} \][/tex]
5. Calculate the heat transferred ([tex]\(Q\)[/tex]):
[tex]\[ Q = 3500 \cdot 4.186 \cdot 5 \][/tex]
[tex]\[ Q = 73255 \, \text{J} \][/tex]
Hence, the amount of heat that must be transferred to 3500 grams of liquid water to change its temperature from [tex]\(27^{\circ}C\)[/tex] to [tex]\(32^{\circ}C\)[/tex] is [tex]\(73255 \, \text{J}\)[/tex].
Comparing this value to the given options:
A. [tex]\(2900 \, \text{J}\)[/tex]
B. [tex]\(73,000 \, \text{J}\)[/tex]
C. [tex]\(170 \, \text{J}\)[/tex]
D. [tex]\(470,000 \, \text{J}\)[/tex]
The closest and most correct option is:
B. [tex]\(73,000 \, \text{J}\)[/tex]
