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 determine the volume of fuel needed to boil 40L of water, we need to calculate the amount of heat required to raise the temperature of the water from 25.0°C to its boiling point (100°C) and then convert that into the amount of fuel needed.
First, let's calculate the amount of heat required to raise the temperature of 40L of water from 25.0°C to 100°C. We can use the formula:
Q = m * c * ΔT
Where:
Q is the amount of heat required
m is the mass of water
c is the specific heat capacity of water (4.184 J/g°C)
ΔT is the change in temperature
Since we know the density of water and the initial volume, we can calculate the mass of water using the formula:
mass = density * volume
mass = 1g/mL * 40L * 1000mL/L = 40,000g
Now, let's calculate the amount of heat required:
Q = 40,000g * 4.184 J/g°C * (100°C - 25.0°C)
Q = 40,000g * 4.184 J/g°C * 75°C
Q = 12,552,000 J
Next, we need to calculate the heat generated from the combustion of the fuel. We can use the equation:
Q = n * ΔH
Where:
Q is the amount of heat generated
n is the number of moles of fuel
ΔH is the standard enthalpy of formation of C7H16 (-224.4 kJ/mol)
To find the number of moles of fuel, we will use the formula:
n = mass / molar mass
molar mass of C7H16 = 7 * (12.01 g/mol) + 16 * (1.01 g/mol) = 100.23 g/mol
n = 40,000g / 100.23 g/mol = 399.63 mol
Now, let's calculate the amount of heat generated:
Q = 399.63 mol * -224.4 kJ/mol
Q = -89,753.57 kJ
Since 15% of the heat generated goes to heating the water, we can calculate the heat actually transferred to the water:
heat_transferred = 0.15 * Q
heat_transferred = 0.15 * -89,753.57 kJ
heat_transferred = -13,463.03 kJ
Finally, we can convert the heat transferred to the amount of fuel needed using the formula:
fuel_volume = heat_transferred / (density * ΔH)
density of the fuel = 0.78 g/mL
ΔH = -224.4 kJ/mol
fuel_volume = -13,463.03 kJ / (0.78 g/mL * -224.4 kJ/mol)
fuel_volume = 21,005.76 mL
To express the answer in two significant figures, we round the fuel_volume to:
fuel_volume = 21,000 mL
Therefore, you should bring 21,000 mL of fuel for your backpacking trip.