When backpacking in the wilderness, hikers often boil water to sterilize it for drinking. Suppose that you are planning a backpacking trip and will need to boil 40L of water for your group. What volume of fuel should you bring? Assume each of the following: the fuel has an average formula of C7H16; 15% of the heat generated from combustion goes to heat the water (the rest is lost to the surroundings); the density of the fuel is 0.78g/mL; the initial temperature of the water is 25.0C and the standard enthalpy of formation of C7H16 is -224.4kJ/mol.
Express your answer using two significant figures.
To calculate the volume of fuel needed to boil 40L of water, we need to consider the energy required to heat the water and the efficiency of the fuel.
Step 1: Calculate the energy required to heat the water.
The specific heat capacity of water is 4.18 J/g°C.
The temperature change needed is from 25.0°C (room temperature) to 100°C (boiling point).
Therefore, the energy required is:
40 L x 1000 g/L x 4.18 J/g°C x (100 - 25)°C = 125,400,000 J
Step 2: Calculate the heat generated from combustion.
The standard enthalpy change of formation of C7H16 is -224.4 kJ/mol.
The molar mass of C7H16 is ((12.01 g/mol x 7) + (1.01 g/mol x 16)) = 100.21 g/mol.
Since the density of the fuel is 0.78 g/mL, we can calculate the volume of the fuel (V_fuel) in mL using the mass of the fuel (m_fuel) in g:
V_fuel = (mass of fuel) / (density of fuel) = (125,400,000 J) / (-224.4 kJ/mol) x (100.21 g/mol) / (0.78 g/mL) = 213, 567 mL
Step 3: Convert the volume of fuel to liters.
213,567 mL x (1 L / 1000 mL) = 213.6 L
Therefore, you need to bring approximately 214L of fuel for your backpacking trip.
To calculate the volume of fuel needed to boil 40L of water, we need to determine the amount of heat required to raise the water temperature from 25.0C to 100C, and then factor in the efficiency of the fuel and its density.
1. Calculate the heat required to raise the temperature of the water:
The specific heat capacity of water is approximately 4.18 J/g°C. Since we have 40L of water, which has a density of 1g/mL, the mass of the water is 40,000g (40L x 1000g/L).
The temperature change is (100 - 25) = 75°C.
The heat required is given by: Q = mass x specific heat x temperature change.
Q = (40,000g) x (4.18 J/g°C) x (75°C)
Q = 12,540,000 J
2. Determine the heat generated from the combustion of the fuel:
Since 15% of the heat generated from combustion goes to heat the water, the actual heat generated is 15% of the total heat required.
Actual Heat Generated = 0.15 x 12,540,000 J
Actual Heat Generated = 1,881,000 J
3. Convert the heat to kilojoules:
1,000 J = 1 kJ
Actual Heat Generated = 1,881,000 J / (1,000 J/kJ)
Actual Heat Generated = 1,881 kJ
4. Calculate the number of moles of C7H16 required to generate this amount of heat:
Using the standard enthalpy of formation of C7H16 (-224.4 kJ/mol), we can calculate the number of moles of fuel needed.
Moles of C7H16 = Actual Heat Generated / Enthalpy of Formation
Moles of C7H16 = 1,881 kJ / (-224.4 kJ/mol)
Moles of C7H16 = -8.38 mol (rounded to two significant figures)
Note: The negative sign arises because the enthalpy of formation is typically reported as a negative value.
5. Calculate the mass of the fuel required using its molar mass:
The molecular formula of C7H16 has a molar mass of 100.2 g/mol.
Mass of C7H16 = Moles of C7H16 x Molar mass
Mass of C7H16 = -8.38 mol x 100.2 g/mol
Mass of C7H16 = -838.68 g (rounded to two significant figures)
Note: The negative sign is a result of the negative moles obtained in the previous step. However, it is not physically meaningful, so we take its absolute value.
6. Convert the mass of the fuel to volume using its density:
The density of the fuel is given as 0.78 g/mL.
Volume of fuel = Mass of C7H16 / Density
Volume of fuel = 838.68 g / 0.78 g/mL
Volume of fuel = 1075.38 mL (rounded to two significant figures)
Final Answer: The volume of fuel you should bring is approximately 1075 mL.