Answer :
To determine the amount of fuel that the backpacker needs to carry in order to heat 2.6 kg of water from 29°C to 100°C, we will follow these steps:
1. Convert the mass of water from kilograms to grams:
[tex]\[ \text{Mass of water} = 2.6 \, \text{kg} \times 1000 \, \text{g/kg} = 2600 \, \text{g} \][/tex]
2. Determine the temperature change (\(\Delta T\)):
[tex]\[ \Delta T = 100^\circ C - 29^\circ C = 71^\circ C \][/tex]
3. Calculate the amount of heat required (Q) using the formula:
[tex]\[ Q = \text{mass} \times \text{specific heat capacity} \times \Delta T \][/tex]
Given:
[tex]\[ \text{mass} = 2600 \, \text{g}, \quad \text{specific heat capacity} = 4.184 \, \text{J/(g·}^\circ C\text{)} \][/tex]
[tex]\[ Q = 2600 \, \text{g} \times 4.184 \, \text{J/(g·}^\circ C\text{)} \times 71^\circ C = 772366.4 \, \text{J} \][/tex]
4. Convert the heat required to kilojoules:
[tex]\[ Q = 772366.4 \, \text{J} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = 772.3664 \, \text{kJ} \][/tex]
5. Determine the amount of fuel needed using the heat per gram of fuel:
[tex]\[ \text{Heat per gram of fuel} = 36 \, \text{kJ/g} \][/tex]
[tex]\[ \text{Fuel needed} = \frac{Q}{\text{Heat per gram of fuel}} = \frac{772.3664 \, \text{kJ}}{36 \, \text{kJ/g}} = 21.454622 \, \text{g} \][/tex]
6. Round the answer to two significant figures:
[tex]\[ \text{Fuel needed} \approx 21 \, \text{g} \][/tex]
So, the backpacker should carry 21 grams of fuel.
1. Convert the mass of water from kilograms to grams:
[tex]\[ \text{Mass of water} = 2.6 \, \text{kg} \times 1000 \, \text{g/kg} = 2600 \, \text{g} \][/tex]
2. Determine the temperature change (\(\Delta T\)):
[tex]\[ \Delta T = 100^\circ C - 29^\circ C = 71^\circ C \][/tex]
3. Calculate the amount of heat required (Q) using the formula:
[tex]\[ Q = \text{mass} \times \text{specific heat capacity} \times \Delta T \][/tex]
Given:
[tex]\[ \text{mass} = 2600 \, \text{g}, \quad \text{specific heat capacity} = 4.184 \, \text{J/(g·}^\circ C\text{)} \][/tex]
[tex]\[ Q = 2600 \, \text{g} \times 4.184 \, \text{J/(g·}^\circ C\text{)} \times 71^\circ C = 772366.4 \, \text{J} \][/tex]
4. Convert the heat required to kilojoules:
[tex]\[ Q = 772366.4 \, \text{J} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = 772.3664 \, \text{kJ} \][/tex]
5. Determine the amount of fuel needed using the heat per gram of fuel:
[tex]\[ \text{Heat per gram of fuel} = 36 \, \text{kJ/g} \][/tex]
[tex]\[ \text{Fuel needed} = \frac{Q}{\text{Heat per gram of fuel}} = \frac{772.3664 \, \text{kJ}}{36 \, \text{kJ/g}} = 21.454622 \, \text{g} \][/tex]
6. Round the answer to two significant figures:
[tex]\[ \text{Fuel needed} \approx 21 \, \text{g} \][/tex]
So, the backpacker should carry 21 grams of fuel.
