Problem 93 - Chemistry: An Atoms-Focused Approach

Question

The density of uranium \(\left(\mathrm{U} ; 19.05 \mathrm{g} / \mathrm{cm}^{3}\right)\) is more than five times as great as that of diamond \(\left(\mathrm{C} ; 3.514 \mathrm{g} / \mathrm{cm}^{3}\right) .\) If you have a cube ( \(1 \mathrm{cm}\) on a side) of each element, which cube contains more atoms?

Step-by-Step Solution

Answer

Consider uranium has a molecular weight of 238.03 g/mol and diamond is made of carbon atoms with a molecular weight of 12.01 g/mol. Answer: The 1 cm³ cube of diamond contains more atoms compared to the 1 cm³ cube of uranium.

Step 1: Find the mass of uranium and diamond cubes

To find the mass of each element in one cubic centimeter (1 cm³), multiply the densities by the volume of the cube (1 cm³): Mass of uranium cube = (19.05 g/cm³) * (1 cm³) = 19.05 g Mass of diamond cube = (3.514 g/cm³) * (1 cm³) = 3.514 g

Step 2: Convert mass to moles

To convert the mass of each element into moles, we need to use their molecular weights: 238.03 g/mol for uranium and 12.01 g/mol for carbon (diamond is made of carbon atoms). Divide the mass by the molecular weight: Moles of uranium = \(\frac{19.05 \thinspace g}{238.03 \thinspace g/mol}\) = 0.08006 mol Moles of diamond = \(\frac{3.514 \thinspace g}{12.01 \thinspace g/mol}\) = 0.2926 mol

Step 3: Calculate the number of atoms in each cube

Now, use Avogadro's number (6.022 x 10²³ atoms/mol) to find the number of atoms in each cube: Atoms in uranium cube = (0.08006 mol) * (6.022 x \(10^{23}\) atoms/mol) = 4.82 x \(10^{22}\) atoms Atoms in diamond cube = (0.2926 mol) * (6.022 x \(10^{23}\) atoms/mol) = 1.76 x \(10^{23}\) atoms

Step 4: Compare the number of atoms

We can now compare the number of atoms in the uranium cube to the number of atoms in the diamond cube. Since 1.76 x \(10^{23}\) atoms (diamond cube) > 4.82 x \(10^{22}\) atoms (uranium cube), the diamond cube contains more atoms.

Key Concepts

Density

Understanding density is key to solving many scientific problems. Density is defined as the mass per unit volume of a substance. It is usually expressed in units of grams per cubic centimeter (g/cm³).

The formula to calculate density is:

• Density = Mass / Volume

For instance, in our exercise, we calculate the density of two elements: uranium and diamond. The density of uranium is given as 19.05 g/cm³, meaning that one cubic centimeter of uranium weighs 19.05 grams. Similarly, diamond has a density of 3.514 g/cm³, indicating that a cubic centimeter of diamond weighs 3.514 grams.

Densities vary significantly between substances. This concept helps us understand why some materials are heavier than others for the same volume, like uranium being denser than diamond.

Molecular Weight

Molecular weight, also known as molecular mass, is the weight of a single molecule of a chemical compound. It is determined by adding up the atomic weights of all atoms in the molecule. Typically, it is expressed in units of grams per mole (g/mol).

In the exercise, the molecular weights are critical for converting mass to moles. For uranium, the molecular weight is given as 238.03 g/mol, and for carbon (the element present in diamonds), it is 12.01 g/mol.

These values allow us to understand how heavy or light a mole of a substance is. By dividing the mass of a substance by its molecular weight, we can find out how many moles are present, which is an essential step for further calculations involving these substances.

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry used for counting entities at the atomic scale. It is defined as the number of atoms, molecules, or other particles in one mole of a substance. Its value is approximately 6.022 x 10²³ particles/mol.

This concept is pivotal in converting between moles and atoms as seen in the exercise, where we used Avogadro's number to find the number of atoms present in the moles of uranium and diamond.

This number is a bridge between the atomic world and the macroscopic amounts we can measure. By multiplying the number of moles by Avogadro's number, you can determine how many individual atoms are present in a sample.

Molar Mass

Molar mass is a term often used interchangeably with molecular weight, but it specifically refers to the mass of one mole of a chemical element or compound, measured in grams per mole (g/mol).

In practical terms, it tells us how much mass of the substance is needed to have exactly one mole. Understanding molar mass is crucial for laboratory work and calculations, as it allows for the conversion of mass into moles, using the formula:

• Number of Moles = Mass / Molar Mass

In the problem, we converted the mass of uranium and diamond into moles using their respective molar masses (238.03 g/mol for uranium and 12.01 g/mol for carbon). This conversion was necessary to then calculate the number of atoms using Avogadro's number.