Question

A stream flows at a rate of \(5.00 \times 10^{4}\) liters per second (L/s) upstream of a manufacturing plant. The plant discharges \(3.50 \times 10^{3} \mathrm{L} / \mathrm{s}\) of water that contains 65.0 \(\mathrm{ppm} \mathrm{HCl}\) into the stream. (See Exercise 135 for definitions.) a. Calculate the stream's total flow rate downstream from this plant. b. Calculate the concentration of \(\mathrm{HCl}\) in ppm downstream from this plant. c. Further downstream, another manufacturing plant diverts \(1.80 \times 10^{4} \mathrm{L} / \mathrm{s}\) of water from the stream for its own use. This plant must first neutralize the acid and does so by adding lime: $$\mathrm{CaO}(s)+2 \mathrm{H}^{+}(a q) \longrightarrow \mathrm{Ca}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(i) $$ What mass of CaO is consumed in an 8.00-h work day by this plant? d. The original stream water contained 10.2 \(\mathrm{ppm} \mathrm{Ca}^{2+}\) . Although no calcium was in the waste water from the first plant, the waste water of the second plant contains \(\mathrm{Ca}^{2+}\) from the neutralization process. If 90.0% of the water used by the second plant is returned to the stream, calculate the concentration of \(\mathrm{Ca}^{2+}\) in ppm downstream of the second plant.

Short answer

a) The total flow rate downstream from the first manufacturing plant is \(5.00 \times 10^{4} \mathrm{L/s} + 3.50 \times 10^{3} \mathrm{L/s} = 5.35 \times 10^{4} \mathrm{L/s}\). b) The concentration of HCl downstream from the first manufacturing plant is 62.1 ppm. c) The mass of CaO consumed in an 8.00-hour workday by the second plant is 2.87 x 10^3 kg. d) The concentration of Ca²⁺ downstream of the second plant is 10.5 ppm.

Step-by-step solution

a. Total Flow Rate Downstream

To calculate the total flow rate downstream from the first manufacturing plant, we simply need to add the flow rate of the stream and the flow rate of the discharged water from the plant. The formula is: Total Flow Rate = Stream Flow Rate + Plant Discharge Flow Rate So the total flow rate would be: Total Flow Rate = \(5.00 \times 10^{4} \mathrm{L/s} + 3.50 \times 10^{3} \mathrm{L/s}\)

b. Concentration of HCl Downstream

To find the concentration of HCl in ppm downstream from the first manufacturing plant, we will use the formula for ppm: ppm = \(\frac{Mass \ of \ Solute}{Mass \ of \ Solution}\) x \(10^{6}\) The mass of HCl from the discharged water is: Mass of HCl = Flow Rate x Concentration x Time We can find the concentration of HCl downstream by dividing the mass of HCl by the total mass of water downstream and then multiplying by \(10^6\).

c. Mass of CaO Consumed in an 8.00-hour Workday

To find the mass of CaO consumed, we first need to find the moles of H⁺ ions that need to be neutralized. We can use the formula: Moles of H⁺ ions = Flow Rate x Concentration x Time Then, we can find the moles of CaO needed using the stoichiometric ratio from the balanced equation. After that, we need to find the mass of CaO consumed by multiplying the moles of CaO with the molar mass of CaO and convert the time from hours to seconds.

d. Concentration of Ca²⁺ Downstream of the Second Plant

To find the concentration of Ca²⁺ downstream from the second plant, we need to find the mass of Ca²⁺ returned to the stream. This can be done using the following steps: 1. Find the mass of Ca²⁺ initially present in the stream water (using ppm = \(\frac{Mass \ of \ Solute}{Mass \ of \ Solution}\) x \(10^{6}\)). 2. Find the mass of Ca²⁺ added to the stream during the neutralization process. This can be done using the stoichiometric ratio and mass of CaO consumed. 3. Calculate the total mass of Ca²⁺ present after the second plant. 4. Find the concentration of Ca²⁺ in ppm downstream of the second plant by dividing the total mass of Ca²⁺ by the mass of the solution downstream and multiplying by \(10^6\).

Key concepts

Flow Rate Calculations

Flow rate is a crucial measurement in many scientific and engineering applications. In this exercise, we are dealing with a stream and a manufacturing plant that discharges water into it. The flow rate tells us how much water is moving through a system per unit of time. It's like measuring the speed of traffic; but instead of cars, we're counting liters of water per second.
To find the total flow rate downstream, we simply add the flow rate of the stream to that of the plant discharge. This is straightforward because

• the two streams of water combine and flow together.
• Think of it as merging two lanes of traffic into one.

For example, if the stream flows at a rate of \(5.00 \times 10^{4} \mathrm{L/s}\) and the plant discharges \(3.50 \times 10^{3} \mathrm{L/s}\), the total flow rate is:
\(5.00 \times 10^{4} + 3.50 \times 10^{3} = 5.35 \times 10^{4} \mathrm{L/s}\).
This calculation helps us understand how different factors contribute to the total output of water in the system.

Concentration in ppm

Concentration is a way to express how much of a particular substance is present in a solution. Parts per million (ppm) is a common measurement used in chemistry to describe low concentrations of substances. It is especially useful for pollutants or trace elements in solutions.
To calculate concentration in ppm, we use the formula: \(\text{ppm} = \frac{\text{Mass of Solute}}{\text{Mass of Solution}} \times 10^{6}\). This represents one part of solute per million parts of solution.
In this scenario, we need to find the concentration of HCl in the water downstream of the plant. The plant adds water containing 65.0 ppm HCl into the stream. The total mass of solute (HCl) can be found by multiplying the flow rate, concentration, and a given time duration.

• The mass of the solution is the sum of the masses of the main stream and the plant discharge.
• By dividing the mass of HCl by the total mass of water, and then multiplying by \(10^6\), we can find the concentration of HCl in ppm downstream.

Stoichiometry

Stoichiometry is a method in chemistry used to calculate the amounts of reactants and products in a chemical reaction. It is based on the balanced chemical equation of the reaction. This is like using a recipe in cooking to find out how much of each ingredient you need.
In the given problem, stoichiometry helps us determine how much lime (CaO) is needed to neutralize the acidic water from the stream that the manufacturing plant is using. The balanced equation is: \[ \mathrm{CaO}(s) + 2 \mathrm{H}^{+}(aq) \rightarrow \mathrm{Ca}^{2+}(aq) + \mathrm{H}_{2}\mathrm{O}(l) \]
This tells us that one mole of calcium oxide reacts with two moles of hydrogen ions. From the flow rate and the concentration of HCl, we can calculate the total moles of H⁺ ions that need to be neutralized.

• Use the stoichiometric ratio to find the moles of CaO needed.
• Multiply the moles of CaO by its molar mass to find the total mass of CaO required.

This information is key for the plant to manage its resources efficiently, ensuring the chemical reaction is complete.

Acid-Base Neutralization

Acid-base neutralization is an important reaction type where an acid reacts with a base to form water and salt. This reaction is often used in industrial processes to render acidic or basic effluents harmless before they can do damage.
In this problem, a manufacturing plant uses lime (CaO) to neutralize the acid (HCl) in water. The reaction converts harmful hydrogen ions into water and calcium ions, making the water safer.

• Neutralization ensures that water diverted for use by the second plant reaches safe levels before it's returned to the stream.
• By knowing the stoichiometry of the reaction, the plant can efficiently calculate how much lime is needed.
• This also prevents any excess CaO from being used, saving cost and resources.

Overall, this process ensures environmental safety and regulatory compliance by minimizing the chemical impact of manufacturing outputs.