Question

Sodium bicarbonate (baking soda) is commonly used to absorb odor. Its solubility is \(9.6 \mathrm{~g} / 100 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}\) at \(30^{\circ} \mathrm{C}\) and \(16 \mathrm{~g} /\) \(100 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}\) at \(60^{\circ} \mathrm{C}\). At \(60^{\circ} \mathrm{C}, 9.2 \mathrm{~g}\) of baking soda are added to \(46 \mathrm{~g}\) of water. (a) Is the resulting mixture homogeneous at \(60^{\circ} \mathrm{C}\) ? If not, how many grams of baking soda are undissolved? (b) The mixture is cooled to \(30^{\circ} \mathrm{C}\). How many more grams of water are needed to make a saturated solution?

Short answer

If not, how many grams of baking soda remain undissolved? How much more water is required to make a saturated solution at 30°C? Answer: The mixture at 60°C is not homogeneous, and 1.84g of baking soda remain undissolved. To make a saturated solution at 30°C, approximately 49.83g of additional water is needed.

Step-by-step solution

Calculate the solubility at 60°C

First, we will calculate the total amount of baking soda that can dissolve in 46g of water at 60°C. Using the solubility given in the exercise: 16g baking soda / 100g water. Multiply the given solubility ratio by the amount of water: Solubility at 60°C = \(\frac{16}{100} \times 46 = 7.36 \mathrm{~g}\)

Determine if the mixture is homogeneous at 60°C and the amount of undissolved baking soda (if any)

The given amount of baking soda is 9.2g. Since 7.36g can dissolve in 46g of water at 60°C, the mixture won't be homogeneous as the amount of baking soda exceeds the solubility limit. To find the amount of undissolved baking soda: Undissolved baking soda = Given amount - Solubility at 60°C = \(9.2 \mathrm{~g} - 7.36 \mathrm{~g} = 1.84 \mathrm{~g}\)

Calculate the solubility at 30°C

Next, we will calculate the total amount of baking soda that can dissolve in 46g of water at 30°C. Using the solubility given in the exercise: 9.6g baking soda / 100g water. Multiply the given solubility ratio by the amount of water: Solubility at 30°C = \(\frac{9.6}{100} \times 46 = 4.416 \mathrm{~g}\)

Calculate how much more water is needed to make a saturated solution at 30°C

Now we need to determine how much more water is needed to dissolve the total amount of baking soda (9.2g) at 30°C. First, we need to find out how many grams of baking soda can dissolve in 1g of water at 30°C: Solubility ratio at 30°C = \(\frac{9.6 \mathrm{~g\, NaHCO}_{3}}{100 \mathrm{~g\, H}_{2}\mathrm{O}} \Rightarrow \frac{9.6}{100} = \frac{9.2}{x} \) Solving for x (additional water needed): x = \(\frac{9.2 \cdot 100}{9.6} \approx 95.83 \mathrm{~g}\) Since 46g of water is already present, subtract the initial amount of water to find how much more water is needed: Additional water needed = Total water needed - Initial water = \(95.83 \mathrm{~g} - 46 \mathrm{~g} = 49.83 \mathrm{~g}\) So, approximately 49.83g of water is needed to make a saturated solution at 30°C.

Key concepts

Chemical Saturation

Understanding chemical saturation is essential when examining the solubility of a substance like sodium bicarbonate (baking soda) in water. Chemical saturation occurs when a solution reaches a point where it cannot dissolve any more of a substance. This limit varies depending on the temperature of the solution. In our exercise, we observed that 9.2g of baking soda was added to 46g of water at 60°C. However, the solubility at 60°C dictates that only 7.36g of baking soda could dissolve fully to create a saturated solution.

The excess baking soda, in this case, 1.84g, remains undissolved, and it is this understanding of chemical saturation which guides us to determine the homogeneity of the mixture and whether additional water is needed upon cooling to achieve a saturated solution again at 30°C.

Homogeneous Mixtures

When looking at homogeneous mixtures, we expect them to have a uniform composition throughout. This uniformity implies that the various components that make up the mixture are indistinguishable from one another. In the context of our initial mixture at 60°C, it was not homogeneous because the amount of baking soda exceeded what the water could dissolve. To be homogeneous, all the baking soda would need to be completely dissolved, which relates directly to the concept of chemical saturation. At a microscopic level, had the mixture been homogeneous, each sample taken from the solution would have contained the same proportion of water and dissolved baking soda.

Temperature Effect on Solubility

The temperature effect on solubility is a crucial factor that influences how much of a substance can be dissolved in a solvent. As seen in the exercise, the solubility of sodium bicarbonate is higher at 60°C than at 30°C. This increase with temperature is common for many solids in liquids; as the temperature rises, so does the amount of solid that can be dissolved. This relationship is due to the increased kinetic energy at higher temperatures, which allows solvent molecules to more effectively break apart the solute particles and integrate them into the solution. When the mixture's temperature is reduced from 60°C to 30°C, the solubility decreases, resulting in the need for additional water to reach saturation again.

Saturated Solution

A saturated solution is the result when the maximum amount of solute has been dissolved into the solvent at a given temperature. Beyond this point, any additional solute will not dissolve and instead will remain separate as excess. This exercise showcased how to determine the amount of solvent needed to achieve a saturated solution after temperature changes affect solubility. We calculated that to resolve the undissolved 1.84g baking soda at 30°C into a saturated solution, an additional 49.83g of water would be necessary. A saturated solution represents a balance between the solute dissolving and crystallizing under a specific set of conditions, and it's a dynamic equilibrium that can shift with changes in temperature or other environmental factors.