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Answer :
To determine how many kilojoules of heat are required to heat 1.37 kg of water from [tex]\( 21.3^{\circ} C \)[/tex] to [tex]\( 89.5^{\circ} C \)[/tex], we can follow these steps:
1. Convert the mass of the water to grams: Since 1 kg equals 1000 grams, we convert 1.37 kg to grams:
[tex]\[
1.37 \text{ kg} \times 1000 \text{ g/kg} = 1370 \text{ g}
\][/tex]
2. Determine the temperature change: This is the difference between the final temperature and the initial temperature:
[tex]\[
89.5^{\circ} C - 21.3^{\circ} C = 68.2^{\circ} C
\][/tex]
3. Calculate the heat required: The specific heat capacity of water is 4.18 J/g°C, meaning it takes 4.18 joules to raise the temperature of 1 gram of water by 1°C. Using the specific heat capacity formula:
[tex]\[
Q = mc\Delta T
\][/tex]
Where:
- [tex]\(Q\)[/tex] is the heat required
- [tex]\(m\)[/tex] is the mass in grams
- [tex]\(c\)[/tex] is the specific heat capacity (4.18 J/g°C for water)
- [tex]\(\Delta T\)[/tex] is the temperature change
Substituting in the values:
[tex]\[
Q = 1370 \text{ g} \times 4.18 \text{ J/g°C} \times 68.2^{\circ} C = 390554.12 \text{ J}
\][/tex]
4. Convert the heat from joules to kilojoules: Since 1 kilojoule equals 1000 joules, convert 390554.12 joules to kilojoules:
[tex]\[
390554.12 \text{ J} \div 1000 = 390.55412 \text{ kJ}
\][/tex]
Therefore, to heat 1.37 kg of water from [tex]\( 21.3^{\circ} C \)[/tex] to [tex]\( 89.5^{\circ} C \)[/tex], 390.55412 kilojoules of heat are required.
1. Convert the mass of the water to grams: Since 1 kg equals 1000 grams, we convert 1.37 kg to grams:
[tex]\[
1.37 \text{ kg} \times 1000 \text{ g/kg} = 1370 \text{ g}
\][/tex]
2. Determine the temperature change: This is the difference between the final temperature and the initial temperature:
[tex]\[
89.5^{\circ} C - 21.3^{\circ} C = 68.2^{\circ} C
\][/tex]
3. Calculate the heat required: The specific heat capacity of water is 4.18 J/g°C, meaning it takes 4.18 joules to raise the temperature of 1 gram of water by 1°C. Using the specific heat capacity formula:
[tex]\[
Q = mc\Delta T
\][/tex]
Where:
- [tex]\(Q\)[/tex] is the heat required
- [tex]\(m\)[/tex] is the mass in grams
- [tex]\(c\)[/tex] is the specific heat capacity (4.18 J/g°C for water)
- [tex]\(\Delta T\)[/tex] is the temperature change
Substituting in the values:
[tex]\[
Q = 1370 \text{ g} \times 4.18 \text{ J/g°C} \times 68.2^{\circ} C = 390554.12 \text{ J}
\][/tex]
4. Convert the heat from joules to kilojoules: Since 1 kilojoule equals 1000 joules, convert 390554.12 joules to kilojoules:
[tex]\[
390554.12 \text{ J} \div 1000 = 390.55412 \text{ kJ}
\][/tex]
Therefore, to heat 1.37 kg of water from [tex]\( 21.3^{\circ} C \)[/tex] to [tex]\( 89.5^{\circ} C \)[/tex], 390.55412 kilojoules of heat are required.
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