Questions: Suppose a boil water notice is sent out advising all residents in the area to boil their water before drinking it or using it for cooking. You need to boil 15.5 L of water using your natural gas (primarily methane) stove. What volume of natural gas is needed to boil the water if only 19.7% of the heat generated goes towards heating the water. Assume the density of methane is 0.668 g / L, the density of water is 1.00 g / mL, and that the water has an initial temperature of 22.2°C. Enthalpy of formation values can be found in this table. Assume that gaseous water is formed in the combustion of methane.

Solution Steps

First, we need to determine the energy required to heat 15.5 L of water from \(22.2^\circ \mathrm{C}\) to \(100^\circ \mathrm{C}\) and then to convert it to steam.

1. Calculate the mass of the water: \[ \text{Mass of water} = 15.5 \, \text{L} \times 1000 \, \text{mL/L} \times 1.00 \, \text{g/mL} = 15500 \, \text{g} \]

2. Calculate the energy required to heat the water to \(100^\circ \mathrm{C}\): \[ q_1 = m \cdot c \cdot \Delta T \] where \(m = 15500 \, \text{g}\), \(c = 4.184 \, \text{J/g}^\circ \mathrm{C}\), and \(\Delta T = 100^\circ \mathrm{C} - 22.2^\circ \mathrm{C} = 77.8^\circ \mathrm{C}\).

   \[ q_1 = 15500 \, \text{g} \times 4.184 \, \text{J/g}^\circ \mathrm{C} \times 77.8^\circ \mathrm{C} = 5,048,000 \, \text{J} \]

3. Calculate the energy required to convert the water to steam: \[ q_2 = m \cdot \Delta H_{\text{vap}} \] where \(\Delta H_{\text{vap}} = 2260 \, \text{J/g}\).

   \[ q_2 = 15500 \, \text{g} \times 2260 \, \text{J/g} = 35,030,000 \, \text{J} \]

4. Total energy required: \[ q_{\text{total}} = q_1 + q_2 = 5,048,000 \, \text{J} + 35,030,000 \, \text{J} = 40,078,000 \, \text{J} \]

1. Determine the energy provided by the combustion of methane: The combustion of methane (\(\mathrm{CH_4}\)) releases \(890.3 \, \text{kJ/mol}\).

2. Convert the energy required to kJ: \[ 40,078,000 \, \text{J} = 40,078 \, \text{kJ} \]

3. Calculate the moles of methane needed: Since only \(19.7\%\) of the heat generated is used to heat the water: \[ \text{Effective energy required} = \frac{40,078 \, \text{kJ}}{0.197} = 203,950 \, \text{kJ} \]

   \[ \text{Moles of methane} = \frac{203,950 \, \text{kJ}}{890.3 \, \text{kJ/mol}} = 229.2 \, \text{mol} \]

1. Convert moles of methane to grams: \[ \text{Molar mass of methane} = 16.04 \, \text{g/mol} \]

   \[ \text{Mass of methane} = 229.2 \, \text{mol} \times 16.04 \, \text{g/mol} = 3677 \, \text{g} \]

2. Convert grams of methane to volume: \[ \text{Density of methane} = 0.668 \, \text{g/L} \]

   \[ \text{Volume of methane} = \frac{3677 \, \text{g}}{0.668 \, \text{g/L}} = 5505 \, \text{L} \]

Final Answer