Question

A student is given a crucible and asked to prove whether it is made of pure platinum. She first weighs the crucible in air and then weighs it suspended in water (density = \(0.9986 \mathrm{~g} / \mathrm{mL}\) ). The readings are \(860.2 \mathrm{~g}\) and \(820.2 \mathrm{~g}\), respectively. Based on these measurements and given that the density of platinum is \(21.45 \mathrm{~g} / \mathrm{cm}^{3},\) what should her conclusion be? (Hint: An object suspended in a fluid is buoyed up by the mass of the fluid displaced by the object. Neglect the buoyancy of air.)

Short answer

The crucible is likely made of pure platinum based on the calculated density of 21.47 g/cm³, close to platinum's density of 21.45 g/cm³.

Step-by-step solution

Understanding the concept of buoyancy

When an object is submerged in a fluid, it experiences an upward force or buoyancy equal to the weight of the fluid displaced by the object. This means the apparent weight of the object in the fluid is less than its true weight in air.

Calculate the buoyant force

The buoyant force acting on the crucible can be calculated using the difference in its weight in air and water. \[ F_b = 860.2 \, \text{g} - 820.2 \, \text{g} = 40.0 \, \text{g} \] This force in grams can be converted into Newtons (since 1 g = 0.00981 N for practical purposes, we return to grams for simplicity in buoyant analysis here). This is effectively the weight of the water displaced by the crucible.

Calculate the volume of the crucible

Since the buoyant force is equal to the weight of the fluid displaced, we can find the volume of the displaced water (and therefore the volume of the crucible) using the density of water:\[ \text{Volume of water displaced} = \frac{40.0 \, \text{g}}{0.9986 \, \text{g/mL}} = 40.06 \, \text{mL} \] Therefore, the volume of the crucible is 40.06 mL, equivalent to 40.06 cm³ because 1 mL = 1 cm³.

Calculate the density of the crucible material

Now, we can find the density of the crucible. Density is mass divided by volume:\[ \text{Density of material} = \frac{860.2 \, \text{g}}{40.06 \, \text{cm}^3} = 21.47 \, \text{g/cm}^3 \]

Compare calculated density with platinum's density

We compare the calculated density of the crucible material (21.47 g/cm³) with the known density of pure platinum (21.45 g/cm³). The values are very close, indicating that the crucible is highly likely to be made of pure platinum, considering experimental precision.

Key concepts

Understanding Buoyancy

Buoyancy is a force that allows objects to float or feel lighter when submerged in a fluid. This force is actually the result of the pressure difference between the top and bottom of the object, due to the fluid. When you place an object in a fluid, like water, it seems to lose weight. This happens because of the buoyant force, which is an upward push equal to the weight of the fluid that the object displaces. For instance, in the given exercise, the crucible showed different weights in air and in water. This difference is the buoyant force helping us determine how much water is displaced, which is crucial for density calculations.

To calculate the buoyant force, you simply subtract the weight of the object in the fluid from its weight in the air. In our example, the forces are calculated as follows:
- Weight in air: 860.2 g
- Weight in water: 820.2 g
Thus, the buoyant force is 40.0 g, the weight of the water displaced.

Steps to Density Calculation

Determining the density of an object involves calculating the ratio of its mass to its volume. This process helps in identifying substances based on how compactly their mass is packed in a given volume. Moving through the calculation involves two primary steps: finding the volume of the displaced fluid and the density thereafter.

First, we calculate the volume of the water displaced by the crucible, which should equal the volume of the crucible. Using the known density of water (0.9986 g/mL), volume is derived by dividing the buoyant force by this density:

• Buoyant Force: 40.0 g
• Density of Water: 0.9986 g/mL

Volume of the crucible's material, therefore, is: \[\text{Volume} = \frac{40.0 \text{ g}}{0.9986 \text{ g/mL}} = 40.06 \text{ mL}\]

Next, we find the density of the crucible using its mass in air (since this gives us true weight):

• Mass of the crucible: 860.2 g
• Volume of the crucible: 40.06 cm³

So, density (\(\rho\)) is:\[\rho = \frac{860.2 \text{ g}}{40.06 \text{ cm}^3} = 21.47 \text{ g/cm}^3\]This value is close to that of pure platinum, helping in verifying its purity.

Ensuring Experimental Precision

Experimental precision refers to how closely repeated measurements or calculations give the same results. It is a key aspect of scientific experiments, enabling us to trust the data gathered during such processes. The precision in this example emphasizes how even slight differences in measurements are important.

To ensure accuracy, carefully monitor the equipment and procedures used. Consistent weighing conditions, like using the same scale for both air and water measurements, prevent errors. Calibration of tools also factors into securing reliable data. In this problem, the extremely close values of the calculated density (21.47 g/cm³) and the expected density of platinum (21.45 g/cm³) suggest minimal experimental error. The small difference is acceptable within a student lab setup.

To further improve precision, consider:

• Repetition of measurements
• Using freshly calibrated equipment
• Minimizing environmental factors, like drafts or table vibrations

Such measures boost confidence in the results and conclusions drawn, confirming whether the crucible material is near-pure platinum.