Question

The solubility of \(\mathrm{NaCl}\) at \(30^{\circ} \mathrm{C}\) is \(36.3 \mathrm{~g}\) per \(100 \mathrm{~g}\) of water. What mass of \(\mathrm{NaCl}\) is dissolved in a saturated solution that contains \(500.0 \mathrm{~g}\) of water?

Short answer

The mass of NaCl dissolved in a saturated solution that contains 500.0 g of water is 181.5 g.

Step-by-step solution

Identify Known Values

In this problem, the known values are the solubility of NaCl, which is 36.3 g in 100 g of water at 30 degrees Celsius, and the mass of water in the solution, which is 500.0 g.

Set Up Proportion

Since solubility indicates a proportional relationship, a proportion can be set up to solve the problem. Let x represent the mass (in grams) of NaCl that will dissolve in 500.0 g of water. Thus, the proportion to solve is: (36.3 g NaCl / 100 g water) = (x g NaCl / 500.0 g water)

Solve the Proportion

To solve the proportion for x, cross multiply and divide: x = (36.3 g NaCl * 500.0 g water) / 100 g water. Simplify to obtain the mass of NaCl in 500 g of water.

Calculate and Express Answer

The mass of NaCl dissolved in 500 g of water is calculated to be 181.5 g.

Key concepts

Understanding NaCl Solubility

When discussing NaCl solubility, we refer to the amount of sodium chloride (table salt) that can be dissolved in a specific quantity of water before the solution becomes saturated. Knowing the solubility of a substance is critical for various applications, including culinary arts, water treatment, and scientific research. At a given temperature, the solubility value is fixed. For example, at 30°C, the maximum solubility of NaCl is 36.3 grams per 100 grams of water.

Let's imagine a real-life scenario to make this more concrete. If you were preparing a dish and needed to ensure your salt solution was at the brink of saturation, understanding that at 30°C, only 36.3 grams of NaCl can be dissolved in 100 grams of water would be vital information. If you go beyond this point, you'll see salt crystals in the bottom of your container—a clear indicator that the solution has become saturated, meaning it cannot hold any more salt in solution.

Saturated Solutions Explained

A saturated solution represents a point of equilibrium where the solvent, like water, has dissolved as much solute, like NaCl, as it possibly can at a given temperature and pressure. Once a solution is saturated, any additional solute will remain undissolved, precipitating out as solid deposits. It's essential to grasp that saturation is not a static state. It can change with temperature—a warmer solution generally holds more solute than a cooler one.

Here's an analogy to help illustrate saturation: Think of a crowded bus. Once the bus has no more room, new passengers cannot board—it's saturated with people. In a similar way, there comes a point when a water 'vehicle' cannot 'carry' any more dissolved NaCl 'passengers'. Adding more salt to a cup of water at maximum saturation would be analogous to trying to cram more people onto that full bus; it just doesn't fit. Whether in chemistry or in the everyday world, the concept of a saturated solution has a wide range of implications.

Proportional Relationships in Chemistry

In chemistry, a proportional relationship points to a consistent, predictable ratio between two quantities. In this exercise, the proportional relationship is established between the mass of NaCl and the mass of water in which it dissolves to form a saturated solution.

Just as in recipes, where you might double all the ingredients to maintain flavor when cooking a larger portion, in solubility problems you proportionally increase the amount of solute based on the solvent's quantity. For example, if you know that 36.3 grams of NaCl dissolve in 100 grams of water, you can deduce that 72.6 grams would be needed for 200 grams of water, maintaining the same proportional relationship. These relationships are fundamental in preparing solutions in laboratories, pharmaceutical industries, and even in everyday cooking. In the textbook problem, understanding this proportional concept is what enables you to predict that exactly 181.5 grams of NaCl would dissolve in 500 grams of water at the specified temperature.