Suppose that you are planning a backpacking trip and will need to boil 40 L 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 you need, we will need to calculate the heat energy required to raise the temperature of the water to its boiling point and then find the amount of fuel needed to produce that much heat energy.
First, let's calculate the amount of heat energy required to heat the water. We know the specific heat capacity of water is approximately 4.18 J/g·°C, and the initial and final temperatures are 25.0°C and 100.0°C respectively.
The mass of water can be calculated using the equation: mass = volume × density. Here, volume = 40 L, and density = 1 g/mL. So, the mass of water is 40 L × 1000 mL/L × 1 g/mL = 40,000 g.
Next, we calculate the heat energy using the equation: q = m × c × ΔT, where q is the heat energy, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature.
q = 40,000 g × 4.18 J/g·°C × (100.0°C - 25.0°C) = 12,540,000 J.
Since only 15% of the heat generated from combustion goes towards heating the water, we need to divide the calculated heat by 15% (or 0.15) to find the total heat energy generated.
Total heat energy generated = 12,540,000 J / 0.15 = 83,600,000 J.
Now, let's calculate the number of moles of C7H16 in the fuel. The standard enthalpy of formation of C7H16 is -224.4 kJ/mol.
To convert this to J/mol, we multiply by 1000: -224.4 kJ/mol × 1000 J/kJ = -224,400 J/mol.
The number of moles is calculated using the equation: moles = heat energy / enthalpy of formation.
moles of C7H16 = 83,600,000 J / -224,400 J/mol = -372 mol.
Since the volume (V) of the fuel is given in mL, we need to convert moles to milliliters.
To do this, we use the equation: V = moles × molar mass / density, where V is the volume in mL, moles is the number of moles, molar mass is the mass per mole, and density is the density of the fuel.
The molar mass of C7H16 can be calculated by multiplying the molar masses of carbon (12.01 g/mol) and hydrogen (1.008 g/mol) by the number of atoms in C7H16: (12.01 g/mol × 7) + (1.008 g/mol × 16) = 100.27 g/mol.
The volume of the fuel can be calculated as follows:
V = -372 mol × 100.27 g/mol / 0.78 g/mL = -48,000 mL.
Since volume cannot be negative, we take the absolute value:
V = 48,000 mL = 4.8 L.
Therefore, you should bring approximately 4.8 L of fuel for boiling 40 L of water for your backpacking trip.
Note: In this calculation, we assumed that all the available fuel is used to heat the water. In reality, factors such as heat loss to the surroundings and the efficiency of the stove can affect the amount of fuel needed.