Problem 21 - General Chemistry

Question

A 75.0 -g sample of a pure liquid, liquid \(A\), with a density of \(3.00 \mathrm{~g} / \mathrm{mL}\) is mixed with a \(50.0-\mathrm{mL}\) sample of a pure liquid, liquid \(\mathrm{B}\), with a density of \(2.00 \mathrm{~g} / \mathrm{mL}\). What is the total volume of the mixture? (Assume there is no reaction upon the mixing of \(\mathrm{A}\) and \(\mathrm{B}\) and volumes are additive. \()\) a \(275 \mathrm{~mL}\) b \(175 \mathrm{~mL}\) c \(125 \mathrm{~mL}\) d \(100 \mathrm{~mL}\) e \(75 \mathrm{mI}\)

Step-by-Step Solution

Answer

The total volume of the mixture is 75 mL.

Step 1: Calculate Volume of Liquid A

To determine the volume of liquid A, use the formula \( \text{volume} = \frac{\text{mass}}{\text{density}} \). The mass of liquid A is 75.0 g and its density is 3.00 g/mL. Plugging in the values, we get \( \text{volume of A} = \frac{75.0}{3.00} = 25.0 \, \text{mL} \).

Step 2: Identify Volume of Liquid B

The problem states that the volume of liquid B is directly given as \( 50.0 \, \text{mL} \). No calculations are needed for this step since it is provided.

Step 3: Calculate Total Volume of the Mixture

Since volumes are additive and there is no reaction upon mixing, simply add the volumes of liquid A and liquid B. Thus, the total volume is \( 25.0 \, \text{mL} + 50.0 \, \text{mL} = 75.0 \, \text{mL} \).

Key Concepts

Density

Density is a fundamental concept in science that describes how much mass is contained within a specific volume. It tells us how closely packed the particles are in a given material. The formula to calculate density is:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
This relationship shows that if you know any two of the three quantities—density, mass, or volume—you can find the third. For example, in the context of the problem, liquid A has a density of 3.00 g/mL. This means every milliliter of liquid A weighs 3.00 grams. Density is crucial for understanding how substances interact when mixed, as it affects buoyancy, stability, and the layering of liquids.

• Units of density are typically presented as grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³).
• It helps in identifying materials and checking their purity.

Mass

Mass is the measure of the amount of matter in an object or substance, often measured in grams or kilograms. It is one of the fundamental properties of an object. Mass is different from weight, as it does not change with the location or environment, as opposed to weight, which can vary depending on the gravitational force.
For instance, the problem provides the mass of liquid A as 75.0 g. This amount of mass is crucial for calculating its volume when combined with its density. Knowing the mass of each liquid helps us determine how they will behave when mixed.

• It is a scalar quantity, which means it has magnitude but no direction.
• Conservation of mass principle states that mass is neither created nor destroyed in a closed system.

Volume

Volume refers to the amount of three-dimensional space occupied by a substance, often expressed in liters, milliliters, or cubic meters. The formula for finding the volume when mass and density are known is:
\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
Volume is important for determining how much space a liquid will take up, which is essential in many practical applications. In our case, we calculated the volume of liquid A using the formula provided, resulting in 25.0 mL. The volume of liquid B is directly given as 50.0 mL. Understanding their volumes is vital to solving the original exercise question and deciding on storage requirements.

• Common units include liters (L), milliliters (mL), cubic centimeters (cm³).
• In solids, volume can relate to dimensions, while in liquids and gases, containers are used.

Additive Volume

Additive volume is a concept that implies that when two or more substances are combined, the total volume is simply the sum of the individual volumes. This is only true if there is no chemical reaction or significant interaction between the substances that would change their overall volume when mixed.
In the exercise, the volumes of liquids A and B are additive because they do not react with each other when mixed. Thus, their total volume is calculated by adding: 25.0 mL (from liquid A) and 50.0 mL (from liquid B), giving a total volume of 75.0 mL. This principle simplifies many calculations in chemistry and material science where mixtures are common.

• Most commonly applies to liquids and gases.
• Exceptions occur when substances dissolve or react chemically.