Problem 172 - General Chemistry

Question

A sample of a bright blue mineral was weighed in air, then weighed again while suspended in water. An object is buoyed up by the mass of the fluid displaced by the object. In air, the mineral weighed \(7.35 \mathrm{~g} ;\) in water, it weighed \(5.40 \mathrm{~g}\). The densities of air and water are \(1.205 \mathrm{~g} / \mathrm{L}\) and \(0.9982 \mathrm{~g} / \mathrm{cm}^{3}\), respectively. What is the density of the mineral?

Step-by-Step Solution

Answer

The density of the mineral is approximately 3.76 g/cm³.

Step 1: Calculate Apparent Weight Loss

First, determine the apparent weight loss of the mineral when submerged in water. This is done by subtracting the weight in water from the weight in air: \[ 7.35\, \text{g} - 5.40\, \text{g} = 1.95\, \text{g} \]

Step 2: Understand Buoyant Force

The apparent weight loss is due to the buoyant force, which is equal to the weight of the water displaced by the mineral. This means the volume of the mineral is such that it displaces 1.95 g of water.

Step 3: Calculate Volume of Mineral

Since the density of water is \(0.9982\, \text{g/cm}^3\), you can use the formula for density \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\) to find the volume of the displaced water, which is the same as the volume of the mineral. \[\text{Volume of mineral} = \frac{1.95\, \text{g}}{0.9982\, \text{g/cm}^3} \approx 1.953\, \text{cm}^3\]

Step 4: Calculate Density of Mineral

Now that you have the volume of the mineral, calculate its density using its mass in air and the volume. The density \(\rho\) is given by \[\rho = \frac{\text{Mass}}{\text{Volume}} = \frac{7.35\, \text{g}}{1.953\, \text{cm}^3} \approx 3.76\, \text{g/cm}^3\]

Key Concepts

Buoyant Force Fundamentals

When an object is immersed in a fluid, such as water or air, it experiences an upward force called the buoyant force. This force is the result of the pressure differences at different depths within the fluid. The buoyant force is equal to the weight of the fluid displaced by the object.

In practical terms, when you weigh an object in a fluid, it seems to weigh less than it does in air. This apparent weight loss is precisely due to the buoyant force. In our example exercise, the mineral loses weight from 7.35 g in air to 5.40 g in water. Thus, the buoyant force is 1.95 g, as this is the mass of the water displaced by the mineral

• The formula for buoyant force is: \[ F_b = ext{Weight in air} - ext{Weight in fluid} \]
• For our mineral: \[ F_b = 7.35 ext{ g} - 5.40 ext{ g} = 1.95 ext{ g} \]

Understanding buoyant force helps us determine the volume of the object and also plays a crucial role in determining the object's density.

Volume of Mineral Determination

Once you know the buoyant force exerted on the mineral, you can find out the volume of the mineral. This is because the buoyant force is equivalent to the weight of the water displaced by the mineral, and for each gram of displaced water, there is a corresponding volume in cubic centimeters.

The density of water is approximately 0.9982 g/cm³, which gives a direct relationship between mass and volume: \[ ext{Volume} = \frac{1.95 ext{ g}}{0.9982 ext{ g/cm}^3} \approx 1.953 ext{ cm}^3 \] This calculation tells us that the mineral's volume is about 1.953 cm³.

Here are a few key points to remember:

• Volume can be found using the formula: \[ \text{Volume} = \frac{\text{Mass of displaced fluid}}{\text{Density of fluid}} \]
• In this case, the displaced mass is 1.95 grams of water.
• The calculated volume is essential for finding the mineral's density.

This exact understanding allows you to work through any similar problem involving buoyancy and volume determination.

Density of Water and its Role

The density of water is a pivotal factor when analyzing problems involving buoyancy. In this example, water is used as a reference to determine the volume of our mineral. Water generally has a density of about 1 g/cm³, but more precisely, like in our solution, it is stated as 0.9982 g/cm³.

The density of water influences how much mass corresponds to a particular volume. When an object, such as our mineral, displaces a certain mass of water, it simultaneously displaces a volume due to this density.

Some important aspects to remember include:

• Density is defined by the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
• A precise water density like 0.9982 g/cm³ is often used for more accurate calculations.
• Knowing water's density allows you to convert displaced mass into volume accurately.

Understanding water's density is crucial in determining both the volume of objects and in facilitating the calculation of unknown densities, such as the mineral's in this problem.