Question

A cube of gold that is \(1.00 \mathrm{~cm}\) on a side has a mass of 19.3 g. A single gold atom has a mass of \(197.0\) amu. (a) How many gold atoms are in the cube? (b) From the information given, estimate the diameter in \(\AA\) of a single gold atom. (c) What assumptions did you make in arriving at your answer for part (b)?

Short answer

(a) The number of gold atoms in the cube is found by converting the mass of the cube to amu and dividing by the mass of a single gold atom: \(Number\_of\_Atoms = \frac{19.3\,\text{g}}{197\,\text{g/mol}}\cdot 6.022\times 10^{23}\,\text{atoms/mol} \) (b) To estimate the diameter of a single gold atom, we first calculate the volume of one gold atom and assuming it is spherical, we find its radius and then the diameter: \(Diameter_{\AA} = 2 \cdot \sqrt[3]{\frac{3 \cdot Volume\_of\_One\_Atom}{4\pi}} \cdot 10^{8}\) (c) Assumptions made: 1. Gold atoms are considered spherical in shape and uniformly distributed in the cube. 2. There are only gold atoms in the cube, with no gaps, impurities, or other elements.

Step-by-step solution

Convert mass of the cube from grams to amu

The mass of gold cube is 19.3 g. To convert this mass to amu, we will use the conversion factor 1 mole = 6.022 x 10^23 atoms and 1 mole = 197 g (since one gold atom is 197 amu). This way, we can find the number of atoms in the given mass of the gold cube. \(Mass_{amu} = \frac{19.3\,\text{g}}{197\,\text{g/mol}}\cdot 6.022\times 10^{23}\, \text{atoms/mol}\)

Determine the number of gold atoms

Now we divide the mass of the cube in amu by the mass of a single gold atom to find the number of gold atoms in the cube. \(Number\_of\_Atoms = \frac{Mass_{amu}}{197.0\_ amu}\)

Calculate the volume of gold cube

To get closer to finding the diameter, we need to determine the volume of the gold cube, given its side length which is 1.00 cm. Since, cube_volume = side_length³: \(Cube\_volume = (1.00\,\text{cm})^3 = 1.00\,\text{cm}^3\)

Calculate the volume of a single gold atom

Now we can calculate the volume of a single gold atom, by dividing the volume of the gold cube by the number of gold atoms inside the cube. \(Volume\_of\_One\_Atom = \frac{Cube\_volume}{Number\_of\_Atoms}\)

Estimate the diameter of a single gold atom

Given the volume of one gold atom, we can now approximate its diameter. Since gold atoms might be assumed to be spherical in shape, we can first calculate the radius of the atom using the formula for volume which is \(\frac{4}{3}\cdot\pi\cdot r^3 \). Then, we will multiply the radius by 2 to get the diameter. The result will be in cm. To convert it to Angstroms, multiply the diameter by \(10^{8}\). \(Diameter = 2 r = 2 \cdot \sqrt[3]{\frac{3 \cdot Volume\_of\_One\_Atom}{4\pi}}\) \(Diameter_{\AA} = Diameter \cdot 10^{8}\) Note that since we assumed that gold atoms are spherical which might not be accurate, the diameter value calculated is only an estimation and not the exact value.

Answering part (c)

As for part (c), we made the following assumptions: 1. The gold atoms can be considered spherical in shape and distributed uniformly throughout the cube. 2. There are only gold atoms - no gaps, impurities, or other elements within the cube.

Key concepts

Atomic Mass Unit

The atomic mass unit (amu) is a standard unit of mass that quantifies the mass of an individual atom or molecule. It is an essential concept in chemistry and physics for comparing the masses of different atoms. One amu is defined as one-twelfth of the mass of a carbon-12 atom, which approximately equals to 1.66 x 10^-24 grams. This standardization is crucial because it provides a uniform scale for expressing atomic masses, which are usually very small numbers if expressed in grams.

In this exercise, we relate the mass of a gold atom in amu to the mass of a gold cube. By converting grams to amu, which involves using Avogadro's number, students can understand the relationship between microscopic atomic scale and macroscopic measureables like mass. This helps bridge concepts from atomic theory with laboratory measurements.

Spherical Atoms Assumption

When estimating the properties of atoms, one common and simplified assumption is that atoms are spherical. This is particularly used when trying to calculate features like the diameter or volume of an atom, due to its simpler mathematical treatment.

• Atoms may not perfectly be spheres, but assuming they are simplifies calculations.
• Sphere formulas allow us to estimate atomic diameters and radii using volume we can calculate.

In our problem, we calculate the diameter of a gold atom by assuming it to be a sphere. This assumption aids in using simple geometric formulas to work out atomic dimensions, even though extra caution is needed because this can lead to inaccuracies.

Volume Calculation

Volume calculation is an important step in many physics and chemistry problems, especially when dealing with solid materials and their atomic or molecular components. In this exercise, we begin by calculating the volume of a gold cube. Since it is a cube, its volume is straightforward to calculate using the formula for the volume of a cube: equation: \[ \text{Volume}_{\text{cube}} = \text{side length}^3 \] For the gold cube with a side length of 1.00 cm, its volume would be 1.00 cm³.

After determining the number of gold atoms contained in the entire cube, the exercise progresses to finding the volume occupied by a single gold atom. This is achieved by dividing the total volume by the number of atoms, linking microscopic atomic measures to macroscopic, practical quantities.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry that provides the number of atoms, ions, or molecules in one mole of a substance. Its value is approximately 6.022 x 10^23. This constant is pivotal when converting between atomic mass unit measurements and macroscopic sample masses measured in grams.

• It permits linking the microscopic atomic scale to the macroscopic scale through moles.
• Avogadro's Number bridges the gap between individual atoms or molecules and measurable amounts we work with in the lab.

In the context of the given problem, Avogadro's Number allows us to determine how many atoms are contained within a gold cube, given its mass and the mass of a single gold atom in amu. It is a key step in converting grams to amu, and in subsequently calculating the volume occupied by individual atoms. This helps students grasp the magnitude of atomic quantities and their meaningfulness in real-world scenarios.