Question

Water softeners often replace calcium ions in hard water with sodium ions. Because sodium compounds are soluble, the presence of sodium ions in water does not cause the white, scaly residues caused by calci- um ions. However, calcium is more beneficial to human health than sodium because calcium is a necessary part of to human diet, while high levels of sodium intake are linked to increases in blood pressure. The U.S. Food and Drug Administration (FDA) recommends that adults ingest less than 2.4 \(\mathrm{g}\) of sodium per day. How many liters of softened water, containing a sodium concentration of 0.050\(\%\) sodium by mass, docs a person have to consume to excecd the FDA recom- mendation? (Assume a water density of 1.0 \(\mathrm{g} / \mathrm{mL} .\) )

Short answer

A person would have to consume more than 4.8 liters of softened water to exceed the FDA sodium intake recommendation.

Step-by-step solution

Understand the Recommended Sodium Intake

Firstly, note that the FDA recommends less than 2.4 g of sodium intake per day for an adult.

Calculate the Amount of Sodium in 1 Liter of Softened Water

Calculate the mass of sodium in one liter of softened water. Since the density of water is 1.0 g/mL, 1 liter (which is 1000 mL) will have a mass of 1000 g. With a sodium concentration of 0.050%, this means that 1000 g of water will contain 0.050/100 * 1000 g = 0.5 g of sodium.

Calculate the Number of Liters to Exceed the FDA Recommendation

Divide the recommended maximum daily sodium intake by the amount of sodium in one liter of softened water. This equals 2.4 g / 0.5 g/liter = 4.8 liters. Therefore, a person would have to consume more than 4.8 liters of softened water to exceed the FDA recommendation.

Key concepts

Calcium vs Sodium in Water

Understanding the differences between calcium and sodium in water is essential for grasping their effects on health and household appliances. Hard water is high in minerals like calcium and magnesium, and while these minerals are not harmful to health, and in fact necessary, they can cause issues in plumbing and appliances due to scale build-up. Water softeners combat this by replacing calcium ions with sodium ions. Sodium ions do not precipitate as solids as calcium ions do, which means they won't cause scaling.

When it comes to health, calcium plays a vital role in building strong bones and teeth, aiding in muscle function, and maintaining a regular heartbeat. Unlike excessive calcium which is often not a health concern, high levels of sodium intake can lead to increased blood pressure and associated health risks. It is a balancing act to maintain the right levels of these two ions for both health and practical purposes.

FDA Sodium Intake Recommendation

The U.S. Food and Drug Administration (FDA) has set guidelines on sodium intake for adults to help prevent the adverse effects of consuming too much sodium. The current recommendation is for adults to consume less than 2.4 grams of sodium per day. Excessive sodium in the diet is linked to high blood pressure, heart disease, and stroke. This guideline serves as an important benchmark for evaluating the healthiness of certain foods and beverages, including softened water. Keeping these recommendations in mind is critical when discussing water softeners and the sodium they add to water.

It's also important to consider the source of sodium in one's diet. While softened water might contribute to sodium intake, the majority often comes from processed and prepared foods. Awareness and control over sodium consumption from various sources are necessary to align with the FDA's recommendations.

Calculating Sodium in Softened Water

To calculate the amount of sodium in softened water, understanding the data provided is crucial. For example, if softened water contains 0.050% sodium by mass, this translates to 0.5 grams of sodium per liter of water, given that water has a density of 1.0 g/mL (or 1000 g/L). With the FDA's daily recommendation of less than 2.4 grams of sodium, one can determine how much softened water would exceed this intake threshold.

The simple calculation involves dividing the FDA's daily limit by the amount of sodium per liter in the softened water. This exemplifies the importance of not only knowing the FDA guidelines but also how to interpret and apply percentages and densities in real-life contexts such as determining safe levels of consumption of treated water. It's a practical application of math that can have direct implications on one's health and lifestyle.