Question

(a) You are given a bottle that contains \(4.59 \mathrm{~cm}^{3}\) of a metallic solid. The total mass of the bottle and solid is \(35.66 \mathrm{~g}\). The empty bottle weighs \(14.23 \mathrm{~g}\). What is the density of the solid? (b) Mercury is traded by the "flask," a unit that has a mass of \(34.5 \mathrm{~kg}\). What is the volume of a flask of mercury if the density of mercury is \(13.5 \mathrm{~g} / \mathrm{mL} ?\) (c) A thief plans to steal a gold sphere with a radius of \(28.9 \mathrm{~cm}\) from a museum. If the gold has a density of \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\) what is the mass of the sphere? [The volume of a sphere is \(\left.V=(4 / 3) \pi r^{3} .\right]\) Is he likely to be able to walk off with it unassisted?

Short answer

(a) The density of the metallic solid is \(4.67\frac{\mathrm{g}}{\mathrm{cm^3}}\). (b) The volume of a flask of mercury is \(2555.56 \mathrm{~mL}\). (c) The mass of the gold sphere is \(1951.63 \mathrm{~kg}\) which is too heavy for the thief to walk off with it unassisted.

Step-by-step solution

Find the mass of the solid

Subtract the mass of the empty bottle from the total mass to find the mass of the solid: \(M_{solid} = M_{total} - M_{bottle}\). Thus, \(M_{solid} = 35.66 \mathrm{~g} - 14.23 \mathrm{~g} = 21.43 \mathrm{~g} \).

Calculate the density

Use the formula for density: \(density = \frac{mass}{volume}\). Thus, \(density = \frac{21.43 \mathrm{~g}}{4.59\mathrm{~cm^3}} = 4.67\frac{\mathrm{g}}{\mathrm{cm^3}}\) (b) Finding the Volume of a Flask of Mercury

Convert the mass of the flask to grams

We are given the mass of the flask in kilograms, but the density is given in grams per milliliter. Convert the mass to grams: \(34.5 \mathrm{~kg} * 1000 \mathrm{~g/kg} = 34500 \mathrm{~g} \)

Calculate the volume

Use the formula for density and rearrange for volume: \(volume = \frac{mass}{density}\). Thus, \(volume = \frac{34500 \mathrm{~g}}{13.5\frac{\mathrm{g}}{\mathrm{mL}}} = 2555.56 \mathrm{~mL}\) (c) Finding the Mass of the Gold Sphere

Calculate the volume of the sphere

Use the formula for the volume of a sphere: \(V = \frac{4}{3} * \pi * r^3\). Thus, \(V = \frac{4}{3} * \pi * (28.9\mathrm{~cm})^3 = 101120.84 \mathrm{~cm^3} \)

Calculate the mass of the sphere

Use the formula for density and rearrange for mass: \(mass = density * volume\). Thus, \(mass = 19.3 \frac{\mathrm{g}}{\mathrm{cm^3}} * 101120.84 \mathrm{~cm^3} = 1951628.02 \mathrm{~g}\) Now, convert this mass to kilograms: \(\frac{1951628.02 \mathrm{~g}}{1000 \mathrm{~g/kg}} = 1951.63 \mathrm{~kg}\)

Determine whether the thief can walk off unassisted

Given that the gold sphere weighs 1951.63 kg, it would be nearly impossible for a person to walk off with it unassisted.

Key concepts

Understanding Mass and Volume Relationship

The mass and volume of a substance are two fundamental properties that are intricately linked through density. Mass is the amount of matter in an object and is usually measured in grams (g) or kilograms (kg), while volume is the space that an object occupies, measured in units such as cubic centimeters (\text{cm}^3) or milliliters (mL).

Density, the ratio of mass to volume, serves as the bridge between these two properties: \( density = \frac{mass}{volume} \). To find the density of an object, you divide its mass by its volume. This concept is crucial for various applications, including determining material purity and designing objects to float or sink. In an educational context, problems like those asking to find the density of a metallic solid in a bottle illustrate the practical use of this relationship. The solution requires you to subtract the empty bottle's mass from the total mass, leaving you with the mass of the solid, and then divide by the solid's volume to find the density.

Converting Units in Chemistry

Chemical calculations often require unit conversions to ensure that measurements are in the correct units for calculations. This step is essential because units like mass and volume can be expressed in multiple forms — such as milliliters to liters for volume or grams to kilograms for mass. Understanding how to convert units is key to solving chemistry problems accurately.

For example, to calculate the volume of a flask of mercury, you need to convert its mass from kilograms to grams because the density is given in grams per milliliter. This is achieved by using the conversion factor \( 1\, kg = 1000\, g \). Similarly, after finding the volume in milliliters, you might need to convert it to liters by using the conversion factor \( 1\, L = 1000\, mL \). These conversions are integral to performing density calculations correctly and are often a necessary step before applying formulas.

Calculating Volume of a Sphere

To calculate the volume of a sphere, we use the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. This formula is derived from the integral calculus but is used in basic mathematics and various fields, such as physics and engineering, to determine the space inside a spherical object.

To determine the volume of a gold sphere, you first cube the radius and then multiply it by \( \pi \) and \( \frac{4}{3} \), which gives you the volume in cubic centimeters. Knowing the volume, and with the given density, you can then calculate the mass of the sphere. This application of volume calculation explains how properties of a sphere, such as its size and material, can translate into understanding its weight and the feasibility of movement, such as a thief attempting to steal a large gold sphere from a museum. The substantial mass result clearly suggests that the endeavor would require significant logistical planning.