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 \(\left.=0.9986 \mathrm{~g} / \mathrm{cm}^{3}\right)\). The readings are \(860.2 \mathrm{~g}\) and \(820.2 \mathrm{~g}\), respectively. Given that the density of platinum is \(21.45 \mathrm{~g} / \mathrm{cm}^{3},\) what should her conclusion be based on these measurements? (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.)

Understanding how to determine the density of objects requires knowledge of Archimedes' principle. This principle states that an object submerged in a fluid experiences an upward force called buoyancy. The magnitude of this buoyant force is equal to the weight of the fluid displaced by the object. Hence, when you weigh an object in a fluid, it appears to be lighter than its actual weight because the buoyant force counteracts some of the object's weight.

When solving our given exercise, it's crucial to apply Archimedes' principle correctly. By using the measurements of the crucible's weights in air and water, one can find the buoyant force exerted by the water. This buoyant force—effectively the weight of water displaced—gives us the volume of the crucible, as the force is equal to the mass of the water displaced multiplied by the acceleration due to gravity, which is a known constant. Once we have the volume, we can proceed to determine the density of the crucible material.

In simple terms, buoyancy is why stuff floats. It's a force that all fluids exert on objects placed in them. The direction of the buoyant force is always upwards against the force of gravity, making the object feel lighter when submerged. The greater the volume of fluid displaced, the greater the buoyant force. This principle explains how ships made of steel, a material much denser than water, can float; their shape displaces a volume of water (and thus a weight) that's equal to or greater than the weight of the ship itself.

In the context of our textbook solution, the concept of buoyancy is used to measure the volume of the crucible by seeing how much water it displaces. This measurement of volume is then used in conjunction with the mass of the crucible to calculate its density.

Density is a fundamental concept in chemistry and physics, defining how much mass is contained within a unit volume of a material. Its universal formula is expressed as \(D = \frac{M}{V}\), with \(D\) representing density, \(M\) representing mass, and \(V\) representing volume. Different materials have characteristic densities—this uniqueness allows us to use density as an identifying property.

In the exercise, we are challenged to prove the composition of the crucible by comparing its density to the known density of platinum. This process is a common method of material verification, especially with precious metals. If the calculated density matches or is extremely close to the known density of platinum, we can infer the crucible is indeed pure platinum. Any significant discrepancy would suggest a different material or a mix of substances.