Problem 62 - General Chemistry: Principles and Modern Applications

Question

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

Answer

The density of the calcite sample is \(0.9970 \text{g/mL}\).

Step 1: 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.

Step 2: 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., \(2.9817 \text{g} / 2.9919 \text{mL} = 0.9970 \text{g/mL}\).

Step 3: 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 \(0.9970 \text{g/mL}\).

Key Concepts

Geology

Geology is the scientific study of Earth, its materials, and the processes acting upon them. Minerals are naturally occurring substances formed through these geological processes. They have a unique and distinct chemical composition and structure, which define their physical properties. Mineralogists, a type of geologist, study these minerals to understand their formation, their properties, and how they can be used. Studying minerals is vital because they make up the rocks that form the Earth’s crust. They are fundamental in the study of geology as they give clues about the Earth’s history and the processes that formed it. By examining minerals, geologists can predict volcanic eruptions, locate oil reserves, and understand environmental changes. Minerals like calcite are scrutinized for their density, which can help us understand their identification and use in various industries.

Mineral Density

Mineral density is a crucial property that helps determine the type of mineral and its purity. Density is defined as mass per unit volume, often expressed in grams per milliliter \(g/mL\). It indicates how compact a substance is. High-density minerals are more closely packed with matter compared to low-density minerals.The density of a mineral is important in identification since each mineral has a characteristic density. For instance, by knowing the densities of bromoform and chloroform, we can determine the density of a mineral like calcite, which is key in identifying it. Mineral density not only helps in identification but also in understanding a mineral's composition.In practical applications, density can dictate the usage of a mineral. In industries, higher-density minerals might be preferred for their durability or resistance to impact. Understanding mineral density helps both in geological explorations and in economic applications of various minerals.

Liquids Mixture Technique

The liquids mixture technique is a method used to determine the density of a mineral by creating a mixture of liquids in which the mineral just floats. This specific floatation occurs when the mineral’s density matches that of the liquid mixture, making it an effective way to measure mineral density.In the provided example, a mixture of bromoform and chloroform is used. Bromoform \(d=2.890 \text{g/mL}\) and chloroform \(d=1.444 \text{g/mL}\) can be mixed in varying proportions. When these liquids are mixed, the density can be adjusted so that the calcite mineral grains neither float nor sink, indicating that their densities are equal.This technique is particularly useful in geology for measuring mineral densities without the need for complex equipment. It provides an accurate measure through a relatively simple process, making it ideal for field studies where precision is required with minimal resources. This method helps bridge a gap between theoretical learning and practical applications, giving students and professionals a hands-on approach to understanding mineral properties.