A chemist mixes two liquids A and B to form a homogeneous...

A chemist mixes two liquids $A$ and $B$ to form a homogeneous mixture. The densities of the liquids are $2.0514 \mathrm{g} / \mathrm{mL}$ for $\mathrm{A}$ and $2.6678 \mathrm{g} / \mathrm{mL}$ for $\mathrm{B}$. When she drops a small object into the mixture, she finds that the object becomes suspended in the liquid; that is, it neither sinks nor floats. If the mixture is made of 41.37 percent $A$ and 58.63 percent $B$ by volume, what is the density of the object? Can this procedure be used in general to determine the densities of solids? What assumptions must be made in applying this method?

Key Concepts

Assumptions in Density Determination via Buoyant Suspension

Using the flotation method to measure density relies on several assumptions: the mixed liquids are completely miscible and form a homogeneous solution, the mixing does not lead to significant volume contraction or expansion, and there is no interaction (such as chemical reaction or absorption) between the object and the liquids beyond the buoyant force. Furthermore, it assumes that temperature and other environmental factors that might affect density are controlled or negligible.

Archimedes' Principle and Buoyancy

Archimedes' Principle states that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. When an object is suspended in a fluid—neither sinking nor floating—it indicates that its density is equal to that of the surrounding fluid. This principle underpins the experimental method used to determine the density of an object by matching it to a fluid of known density.

Weighted Average in Mixture Density

When mixing two liquids, the density of the resulting homogeneous mixture can be computed as the weighted average of the densities of the individual liquids, where each component's contribution is proportional to its volume fraction. This approach assumes that the volumes add directly without significant contraction or expansion.

Density (Mass per Unit Volume)

Density is a measure of how much mass is contained in a unit volume. In general, it is calculated as mass divided by volume. For mixtures, the overall density is determined by combining the densities of the individual components according to their proportions.

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