Question

You have some metal shot (small spheres like BBs), and you want to identify the metal. You have a flask that is known to contain exactly \(100.0 \mathrm{~mL}\) when filled with liquid to a mark in the flask's neck. When the flask is filled with water at \(20^{\circ} \mathrm{C}\), the mass of flask and water is \(122.3 \mathrm{~g}\). The water is emptied from the flask and 20 of the small spheres of metal are carefully placed in the flask. The 20 small spheres had a mass of \(42.3 \mathrm{~g}\). The flask is again filled to the mark with water at \(20^{\circ} \mathrm{C}\) and weighed. This time the mass is \(159.9 \mathrm{~g}\). (a) What metal is in the spheres? (Assume that the spheres are all the same and consist of pure metal.) (b) Calculate the volume occupied by 500 spheres.

Short answer

The metal is likely tin. Volume of 500 spheres is 940 mL.

Step-by-step solution

Calculate the Density of the Metal

First, we need to find the density of the metal. The density \( \rho \) is given by the formula \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume. We have the mass of the 20 spheres as \( 42.3 \, \text{g} \).The volume of water displaced by the spheres will help us find the volume of the spheres because the flask with spheres and water weighs \( 159.9 \, \text{g} \) while the empty flask weighs \( 122.3 \, \text{g} \) without spheres. The difference is due to water displaced by spheres: \( 159.9 \, \text{g} - 122.3 \, \text{g} = 37.6 \, \text{g} \) of water displaced. Since 1 mL of water weighs 1 g, the volume displaced is \( 37.6 \, \text{mL} \). Thus, the volume of the 20 spheres is \( 37.6 \, \text{mL} \), so the volume for one sphere will be \( 1.88 \, \text{mL} \).Now calculate the density: \[ \rho = \frac{42.3 \, \text{g}}{37.6 \, \text{mL}} = 2.25 \, \text{g/mL} \].

Identify the Metal

Compare the calculated density to known densities of metals. Common metals close to this density value include silver, manganese, and tin. Tin has an approximate density of \( 2.25 \, \text{g/mL} \), matching our calculated density, thus the metal is likely tin.

Calculate the Volume Occupied by 500 Spheres

We've determined one sphere has a volume of \( 1.88 \, \text{mL} \). For 500 spheres, multiply the volume of one sphere by 500:Volume of 500 spheres \[ 500 \times 1.88 \, \text{mL} = 940 \, \text{mL} \].

Key concepts

Volume Displacement

When you place an object in a fluid, it pushes or "displaces" some of that fluid out of the way. This displacement can help us determine the volume of objects, which is particularly useful for irregularly shaped or small items, like metal spheres. In our example, when placing 20 metal spheres into a flask, we noticed the increase in total weight due to water displacement.
- Initially, the flask weighed 122.3 grams with water.
- When the spheres were added, the weight increased to 159.9 grams.
So, the spheres displaced 37.6 grams (or milliliters) of water, because the mass difference corresponds directly to the displaced volume when dealing with water (1 gram = 1 mL).
Thus, the displaced water gives us the total volume of these 20 metal spheres. Volume displacement is a handy method because it delivers direct volume measurements without the necessity to mathematically estimate or measure an object's size using geometric means.

Metal Identification

Identifying a metal based on physical properties is a process of elimination. In this exercise, we focused on density—a key property unique to each metal. Knowing the density, we can compare it against standard values for different metals, and find a match.
Once we calculated the density of our metal spheres as 2.25 g/mL, the task was to compare this with known densities:

• Silver has a density close to 10.49 g/mL
• Manganese's density is around 7.2 g/mL
• Tin's density is approximately 2.25 g/mL

The nearest match was tin, aligning perfectly with our calculated density value. Thus, we deduced that the metal in question is likely tin. This practical approach simplifies metal identification by narrowing down potential matches using density tables.

Density Calculation

Density is a measure of how much mass is contained in a given volume. The formula to find density is simple: \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume.
In the exercise, you have the total mass of 20 spheres, which is 42.3 grams, and you've already calculated the total volume displaced by these spheres as 37.6 mL. Plugging these values into the formula gives you the density:
\[ \rho = \frac{42.3 \, \text{g}}{37.6 \, \text{mL}} = 2.25 \, \text{g/mL} \]
Understanding how density works not only helps in identifying a material but also aids in predicting how the material will interact with its surroundings. A higher density means the substance is heavier for its size, and this property is crucial in many practical applications, like buoyancy and material strength.