Problem 46 - Chemistry and Chemical Reactivity

Question

You want to determine the density of a compound but have only a tiny crystal, and it would be difficult to measure mass and volume accurately. There is another way to determine density, however, called the flotation method. If you placed the crystal in a liquid whose density is precisely that of the substance, it would be suspended in the liquid, neither sinking to the bottom of the beaker nor floating to the surface. However, for such an experiment you would need to have a liquid with the precise density of the crystal. You can accomplish this by mixing two liquids of different densities to create a liquid having the desired density. (a) Consider the following: you mix \(10.0 \mathrm{mL}\) of \(\mathrm{CHCl}_{3}\) \((d=1.492 \mathrm{g} / \mathrm{mL})\) and \(5.0 \mathrm{mL}\) of \(\mathrm{CHBr}_{3}(d=2.890\) \(\mathrm{g} / \mathrm{mL}),\) giving \(15.0 \mathrm{mL}\) of solution. What is the density of this mixture? (b) Suppose now that you wanted to determine the density of a small yellow crystal to confirm that it is sulfur. From the literature, you know that sulfur has a density of \(2.07 \mathrm{g} / \mathrm{cm}^{3} .\) How would you prepare \(20.0 \mathrm{mL}\) of the liquid mixture having that density from pure samples of \(\mathrm{CHCl}_{3}\) and \(\mathrm{CHBr}_{3} ? \quad\) (Note: \(1 \mathrm{mL}=1 \mathrm{cm}^{3}\).)

Step-by-Step Solution

Answer

Density of first mixture is 1.958 g/mL. For 2.07 g/cm³ density, use 11.73 mL of CHCl₃ and 8.27 mL of CHBr₃.

Step 1: Calculate the Mass of Each Liquid

First, calculate the mass of each liquid in the initial mixture. Use the formula: \( ext{Mass} = ext{Volume} \times ext{Density} \). For \( \text{CHCl}_3 \) with \( 10.0 \text{ mL} \), the mass is \( 10.0 \text{ mL} \times 1.492 \text{ g/mL} = 14.92 \text{ g} \). For \( \text{CHBr}_3 \) with \( 5.0 \text{ mL} \), the mass is \( 5.0 \text{ mL} \times 2.890 \text{ g/mL} = 14.45 \text{ g} \).

Step 2: Total Mass of the Mixture

Add the masses of the two liquids to find the total mass of the mixture: \( 14.92 \text{ g} + 14.45 \text{ g} = 29.37 \text{ g} \).

Step 3: Calculate Density of the Mixture

Use the formula for density \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). The total volume of the mixture is \( 15.0 \text{ mL} \). Therefore, the density is \( \frac{29.37 \text{ g}}{15.0 \text{ mL}} = 1.958 \text{ g/mL} \).

Step 4: Setup Equation for Desired Mixture

Next, we need to prepare a \( 20.0 \text{ mL} \) mixture with a density of \( 2.07 \text{ g/cm}^3 \). Let \( V_1 \) be the volume of \( \text{CHCl}_3 \) and \( V_2 \) be the volume of \( \text{CHBr}_3 \) such that \( V_1 + V_2 = 20.0 \text{ mL} \). The target density formula is \( 2.07 = \frac{1.492V_1 + 2.890V_2}{20.0} \).

Step 5: System of Equations

From step 4, we have two equations: \( V_1 + V_2 = 20.0 \) and \( 2.07 \times 20.0 = 1.492V_1 + 2.890V_2 \). Transform the first equation to express one variable, \( V_1 = 20.0 - V_2 \). Substitute into the second equation to solve for \( V_2 \).

Step 6: Solve for Volume of Each Liquid

After substitution, solve the equation \( 41.4 = 1.492(20.0 - V_2) + 2.890V_2 \). Simplify and solve: \( 41.4 = 29.84 - 1.492V_2 + 2.890V_2 \), \( 1.398V_2 = 11.56 \), \( V_2 = \frac{11.56}{1.398} \approx 8.27 \text{ mL} \). For \( V_1, \) compute \( V_1 = 20.0 - 8.27 = 11.73 \text{ mL} \).

Step 7: Conclusion on Mixture Preparation

To have a mixture of density \( 2.07 \text{ g/cm}^3 \), mix approximately \( 11.73 \text{ mL} \) of \( \text{CHCl}_3 \) and \( 8.27 \text{ mL} \) of \( \text{CHBr}_3 \).

Key Concepts

Flotation method

The flotation method is a fascinating technique used to determine the density of a small sample that is difficult to measure directly. Imagine you have a crystal so tiny that measuring its mass and volume accurately poses a challenge. Here is where the flotation method comes to the rescue.

The core idea is to immerse the crystal in a liquid whose density matches exactly that of the crystal. If done correctly, the crystal will neither sink to the bottom nor float to the top—it will hover in the liquid. This "floating" state indicates that the densities are equal.

To achieve this result, you generally need to mix two different liquids whose densities differ, tweaking the proportions until the mixture's density matches the desired value. This is why it's crucial to understand the density and how different components can be combined to reach a specific density.

Density formula

Density is a fundamental concept in physics and chemistry. It's a measure of how much mass is contained in a specific volume. The formula for density is straightforward and given by:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

This formula tells us that density is directly related to mass and inversely related to volume. Applying this formula can guide you when you need to calculate the density of any material. For example, if you have a mass of 10 grams and a volume of 5 milliliters, the density would be calculated as follows:
\[ \text{Density} = \frac{10 \text{ g}}{5 \text{ mL}} = 2 \text{ g/mL} \]

Using this simple formula, you can determine whether a certain material will float or sink when placed in another substance.

Mixture density calculation

Calculating the density of a mixture involves understanding how the masses and volumes of its components contribute to the overall density. When two liquids are mixed, each contributes its own mass and volume, which collectively define the density of the resulting solution.

For instance, if you mix 10 mL of a liquid with density 1.492 g/mL with 5 mL of another liquid with density 2.890 g/mL, here's what you have to do:

• Calculate the mass of each component:
• For the first liquid: \( 10 \text{ mL} \times 1.492 \text{ g/mL} = 14.92 \text{ g} \)
• For the second liquid: \( 5 \text{ mL} \times 2.890 \text{ g/mL} = 14.45 \text{ g} \)

Add these masses to get the total mass: 14.92 g + 14.45 g = 29.37 g. Then, divide by the total volume, 15 mL, to find the density of the mixture:
\[ \text{Density} = \frac{29.37 \text{ g}}{15 \text{ mL}} = 1.958 \text{ g/mL} \]

Understanding this process helps in creating solutions of desired densities for experiments like the flotation method.

Volume and mass relationship

The relationship between volume and mass is crucial when determining or predicting the density of a substance. By definition, density links these two properties, offering insights into how much material is present in a given space.

When you know the mass and volume of a substance, you can easily calculate its density using the density formula. However, if you need to determine one property given the other two (volume, mass, or density), you can rearrange the formula to solve for the unknown.

This relationship is especially significant when dealing with mixtures because by knowing the mass and desired density, you can find the necessary volume and vice versa. For example, when creating a specific-density mixture from two components, you need to balance their volumes carefully to achieve the desired result. Understanding this balance between mass and volume helps in precise laboratory work and experimentation.

Such calculations are common in chemistry, showcasing the vital role density plays in scientific analysis.