Question

In an experiment, a sphere of crystalline sodium chloride \((\mathrm{NaCl})\) was suspended in a stirred tank filled with water at \(20^{\circ} \mathrm{C}\). Its initial mass was \(100 \mathrm{~g}\). In 10 minutes, the mass of sphere was found to have decreased by 10 percent. The density of \(\mathrm{NaCl}\) is \(2160 \mathrm{~kg} / \mathrm{m}^{3}\). Its solubility in water at \(20^{\circ} \mathrm{C}\) is \(320 \mathrm{~kg} / \mathrm{m}^{3}\). Use these results to obatin an average value for the mass transfer coefficient.

Short answer

Question: Calculate the average mass transfer coefficient for the dissolution of NaCl in water at 20°C, given that the initial mass of the NaCl sphere is 100 g, the mass decreases by 10 percent after 10 minutes, the NaCl density is 2160 kg/m³, and its solubility in water at 20°C is 320 kg/m³. Answer: The average mass transfer coefficient for the dissolution of NaCl in water at 20°C is approximately 9.68 × 10^(-6) m/s.

Step-by-step solution

Calculate the mass decrease after 10 minutes

First, we need to determine the mass decrease after 10 minutes. We are given that the mass decreases by 10 percent after 10 minutes. Thus, the mass decrease is: Mass decrease = Initial mass × 0.1 = 100 g × 0.1 = 10 g

Calculate the mass transfer rate

Next, we need to find the mass transfer rate (MTR) by dividing the mass decrease by the time elapsed during dissolution. We are given that 10 minutes have passed, so we convert that into seconds to have the time in SI units. Time = 10 min × 60 s/min = 600 s Mass transfer rate (MTR) = Mass decrease / Time MTR = 10 g / 600 s = 0.0167 g/s

Calculate the concentration difference

Now, we want to find the concentration difference (ΔC) between the surface concentration of NaCl and the bulk solution concentration. We know the solubility of NaCl at 20°C is 320 kg/m³ and the density of NaCl is 2160 kg/m³. The surface concentration (C_surface) can be assumed to be equal to the solubility of NaCl: C_surface = 320 kg/m³ = 320,000 g/m³ The bulk solution concentration (C_bulk) is assumed to be 0, since the tank is continuously stirred, and the NaCl concentration is negligible compared to the amount of water. ΔC = C_surface - C_bulk = 320,000 g/m³ - 0 g/m³ = 320,000 g/m³

Calculating mass transfer area

We need to find the area (A) through which the mass transfer is taking place. For a sphere, the surface area is given by the formula: A = 4 × π × (radius)² Given the initial mass and density of sodium chloride, we can determine the initial volume (V) and radius (r) as follows: Initial volume (V) = Mass / Density = 100 g / 2160 kg/m³ = 4.63 × 10^(-5) m³ Radius (r) = (3 × V/(4 × π))^1/3 = (3 × 4.63 × 10^(-5) m³ / (4 × π))^1/3 ≈ 0.0214 m Now, calculate the surface area: A = 4 × π × (0.0214 m)² ≈ 0.00577 m²

Calculate the average mass transfer coefficient

Finally, we can calculate the average mass transfer coefficient (k) using the mass transfer rate, the concentration difference, and the surface area: k = MTR / (A × ΔC) = 0.0167 g/s / (0.00577 m² × 320,000 g/m³) ≈ 9.68 × 10^(-6) m/s The average mass transfer coefficient is approximately 9.68 × 10^(-6) m/s.

Key concepts

Understanding Mass Transfer Rate

Mass transfer rate is a critical concept in chemical engineering and environmental studies, representing the amount of substance that moves through a unit area over a given period. In the provided exercise, we calculated the mass transfer rate (MTR) of sodium chloride (NaCl) as it dissolves in water. The calculation began by determining the mass decrease, which was 10 g over a span of 10 minutes. When we converted the decrease to a rate, we ended with 0.0167 g/s.

This rate of transfer is essential in processes like dissolution, evaporation, or chemical reactions where species are transported from one phase to another. Understanding how to calculate the rate helps students to design and predict the performance of equipment such as reactors, columns, or membranes where mass transfer is a fundamental operation.

To further clarify, consider the analogy of filling a bathtub with water. The flow rate of water from the faucet (analogous to the mass transfer rate) will determine how quickly the bathtub fills (analogous to the concentration of solute in solution). Just as varying the flow rate from the faucet will alter the filling time, changing conditions that affect the mass transfer rate will impact the concentration and rate of dissolution in a system.

Solubility in Water Explained

Solubility in water refers to the maximum amount of a substance that can be dissolved in water at a specific temperature and pressure. In the context of this exercise, we looked at the solubility of sodium chloride at 20°C, which is 320 kg/m³. Solubility plays a key role in various industries and natural processes, as it dictates how substances interact in aqueous solutions.

For instance, when seasoning food, the solubility of salt in water is what allows it to distribute evenly when cooking. In an industrial context, understanding solubility is vital for processes like waste water treatment, where solubility limits can dictate the effectiveness of contaminant removal.

Factors Affecting Solubility

Several factors can affect the solubility of a substance:

• Temperature: As in the case of NaCl, the solubility usually increases with temperature.
• Pressure: Particularly for gases, pressure can significantly impact solubility.
• Molecular structure: Some substances dissolve better in water due to their polarity or ability to form hydrogen bonds.

Mastery of solubility concepts is critical for students studying chemistry, environmental science, and related fields.

Density of Sodium Chloride

Density is an intensive property of matter which is defined as mass per unit volume. The density of sodium chloride (NaCl), as given in the exercise, is 2160 kg/m³. This measurement is crucial when considering the properties of NaCl in various scenarios, such as calculating the mass of a given volume for industrial use or understanding its behavior when mixed with other substances.

Density is a concept utilized across the sciences. In geology, for example, the density of minerals helps to understand their distribution within the Earth. In the culinary world, it's important for measuring ingredients accurately by volume. For engineers, density is vital for material selection and design considerations.

Importance of Density in Mass Transfer

The relationship between density and mass transfer is subtle but important. Density is used to calculate the volume of the substance in question, which in turn impacts the mass transfer area. This is seen in the exercise when determining the surface area of the sodium chloride sphere, an essential step for finding the mass transfer coefficient. For students, grappling with density's role in these calculations is necessary for fields like materials science and chemical engineering.