A technique once used by geologists to measure the density of a mineral is to mix two dense liquids in such proportions that the mineral grains just float. When a sample of the mixture in which the mineral calcite just floats is put in a special density bottle, the weight is 15.4448 g. When empty, the bottle weighs 12.4631 g, and when filled with water, it weighs 13.5441 g. What is the density of the calcite sample? (All measurements were carried out at \(25^{\circ} \mathrm{C}\), and the density of water at \(25^{\circ} \mathrm{C}\) is \(0.9970 \mathrm{g} / \mathrm{mL}\) ). At the left, grains of the mineral calcite float on the surface of the liquid bromoform \((d=2.890 \mathrm{g} / \mathrm{mL})\) At the right, the grains sink to the bottom of liquid chloroform \((d=1.444 \mathrm{g} / \mathrm{mL}) .\) By mixing bromoform and chloroform in just the proportions required so that the grains barely float, the density of the calcite can be determined (Exercise 62).

Step by step solution

01

Understanding the Problem

The problem asks for the density of the calcite sample. The problem provides the densities of water and two other substances (bromoform and chloroform) and weights of various substances, including an empty bottle, a bottle filled with water and a sample of the mixture in which the calcite just floats. An understanding of how these weights and densities can be used to compute the density of the calcite sample is crucial.

02

Calculation of Density of Mixture

Calculate the weight of the mixture by subtracting the weight of the empty bottle from the weight of the bottle with the mixture. This is achieved by the operation \(15.4448 \text{g} - 12.4631 \text{g} = 2.9817 \text{g}\). Calculate the volume of the mixture by comparing its weight in the bottle with the weight of water the bottle could hold. This is calculated using the operation \(2.9817 \text{g} / 0.9970 \text{g/mL} = 2.9919 \text{mL}\). The density of the mixture can then be found by dividing the mass by the volume, i.e., \(2.9817 \text{g} / 2.9919 \text{mL} = 0.9970 \text{g/mL}\).

03

Calculation of Density of Calcite

The density of the calcite sample will be equal to the density of the mixture, as the problem indicates that the sample just floats in the mixture. Hence, the density of the calcite sample is \(0.9970 \text{g/mL}\).

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