Problem 101 - General Chemistry: Principles and Modern Applications

Question

\(\mathrm{MgCl}_{2}\) often occurs in table salt \((\mathrm{NaCl})\) and is responsible for caking of the salt. A 0.5200 g sample of table salt is found to contain \(61.10 \%\) Cl, by mass. What is the \(\% \mathrm{MgCl}_{2}\) in the sample? Why is the precision of this calculation so poor?

Step-by-Step Solution

Answer

The percent of \( \mathrm{MgCl}_{2} \) in the sample is calculated to be 82.5%. The precision of this calculation is poor due to several reasons including averages used for atomic weights, possible alternate sources of Chlorine, and measurement errors.

Step 1: Calculate mass of chlorine

First, calculate the mass of Chlorine (Cl) in the sample based on the given percentage. As it's given that the Cl content is \(61.10\% \), the weight of Cl in the sample would be \(0.6110 \times 0.5200 \, g = 0.3177 \, g \)

Step 2: Calculate mass of \( \mathrm{MgCl}_{2} \)

Now, calculate the mass of \( \mathrm{MgCl}_{2} \) containing this amount of Cl. In \( \mathrm{MgCl}_{2} \), Cl makes up \( (35.45 \times 2)/(24.31 + 35.45 \times 2) \times 100\% = 74.08\% \) of the mass. Hence, the mass of \( \mathrm{MgCl}_{2} \) is \(0.3177 \, g / 0.7408 = 0.429 \, g \).

Step 3: Calculate percent of \( \mathrm{MgCl}_{2} \)

Next, calculate the percent of \( \mathrm{MgCl}_{2} \) in the total sample, which will be \( (0.429 \, g / 0.5200 \, g) \times 100\% = 82.5\% \)

Step 4: Analyze precision

Finally, analyze the precision of the calculation. The percentage of \( \mathrm{MgCl}_{2} \) found is more than 100%, indicating an error. This could be due to several reasons, including: The atomic weights used are averages, not exact values; Chlorine could come from sources other than \( \mathrm{MgCl}_{2} \), and errors in measuring the mass of the Cl and salt. These reasons contribute to the poor precision of this calculation.

Key Concepts

Caking of Table Salt

When you have table salt, it sometimes clumps together in the shaker. This clumping is known as "caking." Basically, it's when the little grains of salt stick together in lumps rather than staying separate and easy to pour. One common culprit for caking is the compound Magnesium Chloride, or \(\text{MgCl}_2\). But why does \(\text{MgCl}_2\) cause this caking?

• MgCl2 is hygroscopic, meaning it absorbs moisture from the air.
• When it pulls in this moisture, the salt grains can become sticky and clump together.
• This makes it harder to shake out of your salt shaker evenly.

This caking phenomenon can be quite annoying in the kitchen, which is why some salt manufacturers add anti-caking agents to keep the salt flowing smoothly.

Mass Percentage Calculation

Chemical formulas can sometimes appear complex, but they hold crucial information about the composition of a substance. Let's look at how to calculate the mass percentage, which tells you how much of one component is in a mixture or compound. In our exercise, you were given the mass percentage of Chlorine \((\text{Cl})\) in a salt sample. Here's a quick breakdown of the steps to do it:

• First, identify the total mass of the sample. Here, it's 0.5200 grams.
• Compute the mass of Cl using the percentage given — 61.10% in this instance. You'll multiply 0.6110 by the total mass (0.5200 g), yielding 0.3177 g of Cl.
• Once you have that, you find the mass of \(\text{MgCl}_2\) that corresponds to this Cl. Because Cl makes up 74.08% of the \(\text{MgCl}_2\) mass, you divide the Cl mass by this percentage \(0.3177 \, g / 0.7408 = 0.429 \, g\).
• Finally, to find the \(\% \text{MgCl}_2\) in the sample, divide the MgCl2 mass by the total sample mass and multiply by 100 to convert it into a percentage format.

Being able to calculate mass percentages is essential for tasks like determining concentrations and ingredient amounts in various applications.

Precision in Chemistry Calculations

Precision in calculations is key to reliable results in chemistry, and it becomes particularly important when working with small measurements like those in your exercise. The precision of a calculation tells you how close repeated measurements are to each other. But occasionally, results can be off, as seen when your \(\% \text{MgCl}_2\) calculated value exceeded 100%, which is logically impossible for a valid percentage. Some factors that affect precision in chemistry include:

• The accuracy of the atomic weights used: these are often average values and might not reflect the exact isotope composition.
• Measurement errors: slight inaccuracies when weighing compounds can lead to significant deviations, especially in small samples.
• Unaccounted variables: surrounding atmospheric conditions or unknown chemical reactions can introduce unforeseen sources of error.

In this particular exercise, the many small sources of error accumulated to produce an imprecise result. Understanding these precision issues helps chemists refine their methods and detect potential problems in their experiments or calculations.