Question

Magnesium sulfate \(\left(\mathrm{MgSO}_{4}\right)\) has a solubility of \(38.9 \mathrm{~g} /\) \(100 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}\) at \(30^{\circ} \mathrm{C}\). A solution is prepared by adding \(9.50 \mathrm{~g}\) of \(\mathrm{MgSO}_{4}\) to \(25.0 \mathrm{~g}\) of water at \(40^{\circ} \mathrm{C}\). A homogeneous mixture is obtained. Is the solution saturated, unsaturated, or supersaturated? One gram of magnesium sulfate is added to the solution cooled to \(30^{\circ} \mathrm{C}\). Would you expect some of the \(\mathrm{MgSO}_{4}\) to precipitate? If \(\mathrm{so},\) how much? If not, how much more \(\mathrm{MgSO}_{4}\) can be added before precipitation takes place?

Short answer

Answer: The solution is unsaturated. You can add approximately 0.22 g more magnesium sulfate before precipitation occurs.

Step-by-step solution

Calculate the solubility of magnesium sulfate in 25.0 g of water at 30°C

We are given the solubility of magnesium sulfate is 38.9 g per 100 g of water at 30°C. We need to find the solubility in 25.0 g of water at this temperature: Solubility in 25.0 g of water = (38.9 g/100 g) * 25.0 g This calculation will give us the solubility of magnesium sulfate in the given amount of water at 30°C.

Compare the given mass of magnesium sulfate to the theoretical solubility

Now, we need to compare the given mass of magnesium sulfate (9.50 g) to the solubility calculated in step 1 to see if the solution is saturated, unsaturated, or supersaturated: - If the given mass of magnesium sulfate is equal to the solubility, the solution is saturated. - If the given mass of magnesium sulfate is less than the solubility, the solution is unsaturated. - If the given mass of magnesium sulfate is more than the solubility, the solution is supersaturated.

Determine the type of the solution

Based on the comparison in step 2, we can determine if the solution is saturated, unsaturated, or supersaturated.

Calculate the amount of magnesium sulfate needed for precipitation

Since the solution is cooled to 30°C, we need to calculate the solubility of magnesium sulfate at this temperature. If the new mass of magnesium sulfate (including the added 1 g) is more than the solubility at 30°C, precipitation will occur. To find out how much magnesium sulfate can be added before precipitation takes place or how much will precipitate, we can subtract the new mass of magnesium sulfate from the solubility at 30°C. This will give us either the amount of magnesium sulfate that can still be added or the amount that will precipitate.

Key concepts

Saturated Solution

When a solution can no longer dissolve any more solute at a given temperature, it's known as a saturated solution. Think of it like a dance floor filled to maximum capacity, with no room for additional dancers.

A saturated solution is at equilibrium, where the rate of dissolution of the solute equals the rate of precipitation. For magnesium sulfate (\r(\r\(\r\mathrm{\r{MgSO}_\r{4}}\r\r\)\r) at 30 degrees Celsius, this balance is achieved when 38.9 grams of the solute has dissolved in 100 grams of water. If you have less than 38.9 grams in 100 grams of water, like the example in the exercise, it's not yet at capacity - meaning it's unsaturated. Once you reach that limit, it's like closing the doors to the dance floor. For example, if you have 9.50 grams of \r\(\r\mathrm{\r{MgSO}_\r{4}}\r\r\) in 25 grams of water, as in the original problem, you can calculate the solubility in 25.0 g of water:

\r\[\r\text{\r{Solubility in 25.0 g of water} } = \r\left(\r\frac{\r{38.9 g}}\r{100 g}\r\r\right) * 25.0 g\r\]
Understanding the concept of a saturated solution is essential because it helps to predict when a solution can't dissolve more solute, which is critical in both understanding chemical reactions and performing precise experiments.

Supersaturated Solution

A supersaturated solution is like an overcrowded dance floor where, against all odds, extra dancers have been squeezed onto it. It's a solution that contains more solute than it would normally hold at a given temperature.

To achieve this precarious state, you'd typically dissolve more solute at a higher temperature and then carefully cool down the solution without disturbing it. If we take magnesium sulfate, for example, and dissolve more than its equilibrium solubility at 30 degrees Celsius, and then manage to cool it down without any solid forming, we've created a supersaturated solution.

This state is unstable and under the right conditions, the excess solute will start to precipitate out until the solution becomes saturated again. It's like playing musical chairs; once the music stops (or in this case, a seed crystal is added, or the solution is disturbed), the extra dancers (solute) will rush to find a place to sit (precipitate out).

Precipitation

Just like rain falling from the sky, precipitation in chemistry refers to the formation of solid particles within a solution. This process occurs when a solution is supersaturated, and it can no longer maintain the excessive solute in the dissolved state.

In the context of our magnesium sulfate example, once the solution's temperature is lowered to 30 degrees Celsius, if the amount of \r\(\r\mathrm{\r{MgSO}_\r{4}}\r\r\) exceeds its solubility at this temperature, it will start to come out of the solution as a solid precipitate. To predict whether precipitation will happen, and if so, how much, we calculate the difference between the total amount of solute and the solubility limit at the given temperature. If you've added a gram to our existing unsaturated solution and it crosses the solubility threshold, precipitation will occur.

Understanding precipitation is critical because it helps us to isolate and purify substances in the laboratory and is also widely used in industry, water treatment, and geochemical processes.