Problem 57 - Chemistry A Molecular Approach

Question

A hard water sample contains \(0.0085 \%\) Ca by mass (in the form of \(\mathrm{Ca}^{2+}\) ions). How much water (in grams) contains \(1.2 \mathrm{~g}\) of Ca? (1.2 g of Ca is the recommended daily allowance of calcium for adults between 19 and 24 years old. \()\)

Step-by-Step Solution

Answer

X = 1.2 g / 0.000085 = 14117.65 grams

Step 1: Convert Percentage to Decimal Form

Convert the given percentage of calcium in the water sample to decimal form by dividing by 100. For example, 0.0085% becomes 0.000085.

Step 2: Set up the Proportion

Using the decimal form of the calcium percentage, set up a proportion to relate calcium mass to the total water mass. If 0.000085 parts of every 1 part of water is calcium, then 1.2 g of calcium corresponds to X g of water.

Step 3: Solve for the Total Water Mass

Solve for X (the total water mass containing 1.2 g of calcium) by dividing the amount of calcium needed by the decimal form of the percentage of calcium. X = (1.2 g calcium) / 0.000085.

Step 4: Calculate and Find the Answer

Perform the division to find the total grams of water needed. X = 1.2 g / 0.000085.

Key Concepts

Hard Water Chemistry

Hard water refers to water that has a high mineral content, usually calcium and magnesium ions. In daily life, this manifests as water that does not lather soap easily and leaves deposits on fixtures and pipes, known as limescale. From a chemistry standpoint, when you dissolve calcium compounds like calcium carbonate or calcium bicarbonate in water, they dissociate to release \(\mathrm{Ca^{2+}}\) ions. The concentration of these calcium ions in the water determines the 'hardness' of the water.

Understanding hard water is crucial when dealing with issues such as the scaling of pipes, the efficiency of soaps and detergents, and nutritional calculations for water consumption. For instance, if someone is looking to meet their daily calcium intake through water, knowledge of the calcium content in their hard water supply would be indispensable.

Converting Percentages to Decimal

When you come across a percentage, it's presenting a value as a fraction of 100. To convert this percentage into a decimal, which is necessary for a variety of arithmetic calculations, you simply divide the percentage by 100. This step diminishes the value by two decimal places to the left. For instance, a percentage like \(0.0085\%\) becomes \(0.000085\) when converted into a decimal.

This conversion is not just a simple math trick; it's foundational for calculations in chemistry, finance, statistics, and more. When you grasp this concept, you unlock the ability to work with ratios and proportions that are imperative for solving real-world problems, such as calculating how much calcium is in a given amount of water, which is a key factor for dieticians and health-conscious individuals.

Mass Proportion Calculation

The concept of mass proportion in chemistry is about relating the mass of one substance to the total mass of the mixture it's a part of. This is often used to figure out how much of a certain chemical is present in a solution or mixture.

In the context of the exercise, we relate the mass of calcium to the mass of water to find how much water you would need to consume to intake a certain amount of calcium. Once you've converted the percentage concentration of calcium to a decimal, you set up a simple proportion: if \(0.000085\) parts of water are calcium, then \(1.2\) grams of calcium would be found in a certain amount of water. Solving this proportion, you discover the total mass of water required to provide the daily recommended allowance of calcium.