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Ch.18 - Chemistry of the Environment
All textbooksBrown 14th EditionCh.18 - Chemistry of the EnvironmentProblem 85
Chapter 18, Problem 85

The main reason that distillation is a costly method for purifying water is the high energy required to heat and vaporize water. (a) Using the density, specific heat, and heat of vaporization of water from Appendix B, calculate the amount of energy required to vaporize 1.00 gal of water beginning with water at 20 °C. (b) If the energy is provided by electricity costing $0.085/kWh, calculate its cost. (c) If distilled water sells in a grocery store for $1.26 per gal, what percentage of the sales price is represented by the cost of the energy?

Verified step by step guidance
1
Step 1: Convert the volume of water from gallons to liters using the conversion factor 1 gallon = 3.78541 liters.
Step 2: Calculate the mass of the water using the density of water (approximately 1 g/mL or 1 kg/L) and the volume in liters.
Step 3: Calculate the energy required to heat the water from 20 °C to 100 °C using the formula q = mcΔT, where m is the mass of the water, c is the specific heat capacity of water (4.18 J/g°C), and ΔT is the temperature change.
Step 4: Calculate the energy required to vaporize the water at 100 °C using the formula q = mL, where m is the mass of the water and L is the heat of vaporization of water (2260 J/g).
Step 5: Add the energy calculated in steps 3 and 4 to find the total energy required, convert this energy from joules to kilowatt-hours (1 kWh = 3.6 x 10^6 J), and then calculate the cost using the given electricity rate of $0.085/kWh. Finally, determine the percentage of the sales price represented by the energy cost by dividing the energy cost by the sales price and multiplying by 100.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heat of Vaporization

The heat of vaporization is the amount of energy required to convert a unit mass of a liquid into vapor without a change in temperature. For water, this value is significant because it indicates how much energy must be supplied to overcome intermolecular forces during the phase change from liquid to gas. Understanding this concept is crucial for calculating the energy needed to vaporize water in the distillation process.
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Heat Capacity

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this value is relatively high, meaning it can absorb a lot of heat before its temperature changes significantly. This property is important in the context of heating water from an initial temperature (20 °C in this case) to its boiling point before vaporization occurs.
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Heat Capacity

Cost of Energy Calculation

Calculating the cost of energy involves determining the total energy consumed and converting that into a monetary value based on the cost per kilowatt-hour (kWh). In this scenario, after calculating the energy required to vaporize the water, one must convert that energy from joules to kilowatt-hours and then multiply by the cost of electricity to find the total cost. This concept is essential for understanding the economic implications of using distillation for water purification.
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Gibbs Free Energy of Reactions
Related Practice
Textbook Question

The concentration of H2O in the stratosphere is about 5 ppm. It undergoes photodissociation according to: H2O(g) → H(g) + OH(g)

(c) The hydroxyl radical, OH, can react with ozone, giving the following reactions:

OH(g) + O3(g) → HO2(g) + O2(g)

HO2(g) + O(g) → OH(g) + O2(g)

What overall reaction results from these two elementary reactions? What is the catalyst in the overall reaction? Explain.

Textbook Question

The following data were collected for the desturction of O3 by H (O3 + H → O2 + OH) at very low concentrations (b) Calculate the rate constant