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 \(\mathrm{CHBr}_{3} ?\left(\text {Note: } 1 \mathrm{mL}=1 \mathrm{cm}^{3} .\right)\)

Short answer

(a) 1.958 g/mL. (b) Mix 11.73 mL of CHCl3 with 8.27 mL of CHBr3.

Step-by-step solution

Calculate the Mass Contribution of Each Liquid

First, find the mass of each liquid in the mixture. The mass of chloroform (\( \text{CHCl}_3 \)) is given by its density times its volume: \( 10.0 \text{ mL} \times 1.492 \text{ g/mL} = 14.92 \text{ g} \). Similarly, the mass of bromoform (\( \text{CHBr}_3 \)) is: \( 5.0 \text{ mL} \times 2.890 \text{ g/mL} = 14.45 \text{ g} \).

Calculate Total Mass and Volume of the Mixture

The total mass of the mixture is the sum of the masses of the two liquids: \( 14.92 \text{ g} + 14.45 \text{ g} = 29.37 \text{ g} \). The total volume of the mixture is the sum of their volumes, which is given as \( 15.0 \text{ mL} \).

Determine the Density of the Mixture

Density is mass divided by volume. For this mixture: \[ \text{Density} = \frac{29.37 \text{ g}}{15.0 \text{ mL}} = 1.958 \text{ g/mL}. \] So, the density of the mixture is \( 1.958 \text{ g/mL} \).

Setup Equation for Target Density Mixture

To find how much of each liquid is needed for a 20.0 mL mixture with a density of \( 2.07 \text{ g/mL} \), let \( V_1 \) be the volume of \( \text{CHCl}_3 \) and \( V_2 \) be the volume of \( \text{CHBr}_3 \). We have \( V_1 + V_2 = 20.0 \). The total mass of the mixture must be \( 2.07 \text{ g/mL} \times 20.0 \text{ mL} = 41.4 \text{ g} \).

Solve System of Equations

We have two equations: 1. \( V_1 + V_2 = 20.0 \) 2. \( 1.492V_1 + 2.890V_2 = 41.4 \).Substitute \( V_2 = 20.0 - V_1 \) into equation 2: \[ 1.492V_1 + 2.890(20.0 - V_1) = 41.4. \]Solve for \( V_1 \):\[ 1.492V_1 + 57.8 - 2.890V_1 = 41.4 \text{ implies } -1.398V_1 = -16.4 \text{ or } V_1 = \frac{16.4}{1.398} = 11.73 \text{ mL}. \]Thus, \( V_2 = 20.0 - 11.73 = 8.27 \text{ mL}. \)

Confirm Mixture for Target Density

Recalculate the total density using the found volumes: Mass of \( \text{CHCl}_3 \): \( 1.492 \times 11.73 = 17.49 \text{ g} \). Mass of \( \text{CHBr}_3 \): \( 2.890 \times 8.27 = 23.89 \text{ g} \). Total mass: \( 17.49 + 23.89 = 41.38 \text{ g} \), very close to target \( 41.4 \text{ g} \). Thus, the densities work and are consistent with our target.

Key concepts

Flotation Method

The flotation method is a clever technique used to determine the density of a solid compound, especially when the sample size is too small for direct measurement. This method relies on the buoyancy principle, where an object will neither sink nor float when immersed in a liquid with the same density.
Here are the key steps in this method:

• Select two liquids with known different densities.
• Mix these liquids to create a new liquid with a density matching your sample.
• Introduce the sample to the mixture and adjust proportions if needed until the sample remains suspended.

This non-invasive technique is especially useful in experimental chemistry as it does not require the direct measurement of mass or volume, which can be challenging in certain situations.

Density of Mixtures

The density of a mixture can be calculated by understanding the contributions from each component liquid. To find the density, you take the total mass of all components and divide it by the total volume.
For example, consider a mixture of liquids A and B with densities and volumes, respectively. The density of the resulting mixture can be calculated using:\[ \text{Density of Mixture} = \frac{\text{Mass of A} + \text{Mass of B}}{\text{Volume of A} + \text{Volume of B}} \]

• First, determine each component's mass using the formula: \( \text{Mass} = \text{Density} \times \text{Volume} \)
• Add these masses to find the total mass.
• Add the volumes of both components to find the total volume.
• Finally, divide total mass by total volume to find the mixture's density.

Understanding the density of mixtures is crucial in many chemical calculations and formulations, helping to ensure accuracy in experiments and manufacturing processes.

Chemical Calculations

Chemical calculations are essential for determining the correct proportions needed to achieve a desired result in a laboratory setting.
These calculations often involve setting up equations to solve for unknowns, such as the volume or mass of components in a solution. This is crucial when preparing a mixture with a specific desired density or when converting units from one form to another.
For instance, in our example, the density target for a specific mixture was achieved by using a system of equations:

• The sum of the mixture components must equal the desired volume.
• An additional equation to find the total mass using known densities.

Using systems of equations, which often involve substitution or elimination methods, allows for precise adjustments and formulations in chemistry.

Experimental Chemistry

Experimental chemistry involves designing and conducting experiments to test hypotheses, discover new reactions, or determine properties like density. This branch of chemistry requires careful planning and execution to ensure reliable results.
Key aspects include:

• Accurate measurement of chemical quantities and conditions.
• Verification of results through repeat experiments and varying conditions.
• Use of various equipment and techniques to gather data, such as the flotation method in density determination.

A critical element of experimental chemistry is troubleshooting and adjusting experimental setups when initial attempts do not yield expected results. It emphasizes practical applications of theories and is fundamental in advancing chemical knowledge and technology.