Problem 82 - Chemistry: The Central Science in SI Units, Global Edition

Question

A \(32.65-g\) sample of a solid is placed in a flask. Toluene, in which the solid is insoluble, is added to the flask so that the total volume of solid and liquid together is \(50.00 \mathrm{~mL}\). The solid and toluene together weigh \(58.58 \mathrm{~g}\). The density of toluene at the temperature of the experiment is \(0.864 \mathrm{~g} / \mathrm{mL}\). What is the density of the solid?

Step-by-Step Solution

Answer

The density of the solid is approximately \(1.63 \mathrm{~g/mL}\).

Step 1: Identify the given information

We are given the following information: - Mass of the solid = \(32.65 \mathrm{~g}\) - Total volume of solid and liquid = \(50.00 \mathrm{~mL}\) - Mass of solid and toluene = \(58.58 \mathrm{~g}\) - Density of toluene = \(0.864 \mathrm{~g/mL}\)

Step 2: Find the mass of toluene

We can find the mass of toluene by subtracting the mass of the solid from the total mass (solid and toluene). Mass of toluene = Mass of solid and toluene - Mass of solid \(Mass_{toluene} = 58.58 \mathrm{~g} - 32.65 \mathrm{~g} = 25.93 \mathrm{~g}\)

Step 3: Find the volume of toluene

To find the volume of toluene, we can use the density of toluene and the mass of toluene. Density = Mass/Volume Therefore, Volume = Mass/Density \(Volume_{toluene} = \frac{25.93 \mathrm{~g}}{0.864 \mathrm{~g/mL}} = 30.00 \mathrm{~mL}\)

Step 4: Find the volume of the solid

Since we now know the volume of toluene, we can find the volume of the solid by subtracting the volume of toluene from the total volume (solid and liquid). Volume of solid = Total volume - Volume of toluene \(Volume_{solid} = 50.00 \mathrm{~mL} - 30.00 \mathrm{~mL} = 20.00 \mathrm{~mL}\)

Step 5: Calculate the density of the solid

Finally, we can calculate the density of the solid using its mass and volume. Density of solid = Mass of solid / Volume of solid \(Density_{solid} = \frac{32.65 \mathrm{~g}}{20.00 \mathrm{~mL}} = 1.63 \mathrm{~g/mL}\) The density of the solid is approximately \(1.63 \mathrm{~g/mL}\).

Key Concepts

Mass and Volume Relationship

The concepts of mass and volume are key when studying physical properties of substances. Mass refers to the amount of matter in an object and is typically measured in grams (g). Volume measures the amount of space an object occupies, commonly expressed in milliliters (mL) or liters (L). These two properties are intrinsically linked when determining density. To solve problems involving mass and volume, it's important to:

• Know the mass of the solid object or liquid you are working with.
• Determine the total combined volume if a substance is dissolved or contained within a liquid.
• Separate individual volumes if given a mixture, where necessary, for accurate density calculations.

In the provided solution, these principles are used efficiently to break down the problem. Once the mass of the solid and the volume of the combined substances were known, the mass of the liquid could be deduced. This simple subtraction provided insight into the volume and density of each component in the mixture.

Density Determination

Density is a key factor that describes how much mass is present in a given volume. It is calculated with the formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This formula is crucial for extracting how tightly matter is packed in a substance. Higher density indicates more mass packed in the same volume compared to a lower density. When calculating density, make sure:

• The mass and volume measurements are of the same object or mixture.
• The units are consistent, such as grams for mass and milliliters for volume.
• All data used in calculations should be from standard conditions if comparability is needed.

In our task, determining the density of the solid involved both finding its volume and mass isolated from the liquid. Only then the solution used the density formula to give the final answer, demonstrating the role of systematic calculation in accurate density determination.

Solid and Liquid Systems

Investigating solid and liquid systems involves understanding the interaction and relationship between different states of matter. Solids have a definite shape and volume whereas liquids fill their container's shape but have a fixed volume. Solutions like our task often require meticulous calculations to separate and analyze these components. Considerations for solving such problems:

• Identify whether the solid is soluble in the liquid to determine if it affects the total volume.
• Apply subtraction or addition methods to isolate individual properties for each state of the matter.
• Use the density values specified for each substance to calculate unknown properties like volume or mass.

In the task given, toluene acts as a liquid medium in which the solid is suspended, allowing the exploration of how the two interact without dissolving. Calculations involve subtracting known values from total measurements to figure out the characteristics of the solid component, perfectly illustrating solid and liquid systems study methods.