Problem 5 - Introductory Chemistry Atoms First

Question

Suppose you have \(0.10\) mole of uranium, \(U\), atoms. (a) How many uranium atoms do you have? (b) What is the mass in grams of this much uranium?

Step-by-Step Solution

Answer

(a) There are \(6.022 \times 10^{22}\) uranium atoms in 0.10 mole. (b) The mass of 0.10 mole of uranium is 23.8 grams.

Step 1: Find the number of uranium atoms

To find the number of uranium atoms in 0.10 mole, we can use Avogadro's number (6.022 x 10^23 atoms/mole). Number of uranium atoms = (0.10 moles) * (6.022 x 10^23 atoms/mole)

Step 2: Calculate the number of uranium atoms

Now we can calculate the number of atoms: Number of uranium atoms = (0.10 moles) * (6.022 x 10^23 atoms/mole) = 6.022 x 10^22 atoms So, there are 6.022 x 10^22 uranium atoms in 0.10 mole.

Step 3: Determine the molar mass of uranium

To find the mass of 0.10 mole of uranium, we need the molar mass of uranium. Looking at the periodic table, we find that the molar mass of uranium (U) is approximately 238 g/mole.

Step 4: Calculate the mass of 0.10 mole of uranium

Now we can calculate the mass of 0.10 mole of uranium using its molar mass: Mass of uranium = (0.10 moles) * (238 g/mole)

Step 5: Calculate the mass in grams of 0.10 mole of uranium

Now we can find the mass: Mass of uranium = (0.10 moles) * (238 g/mole) = 23.8 g So, the mass of 0.10 mole of uranium is 23.8 grams.

Key Concepts

Avogadro's Number

The scale of atoms is so minute that dealing with them individually is impractical, which is where Avogadro's number becomes essential in chemistry. Named after the scientist Amedeo Avogadro, this constant represents the number of particles, typically atoms or molecules, in one mole of any substance. Avogadro's number is approximately equal to \( 6.022 \times 10^{23} \).

When you're given an amount in moles and asked to find out how many atoms you have, like in our uranium example, you simply multiply the number of moles by this constant. In this case, \(0.10\) mole of uranium equates to \(0.10 \times 6.022 \times 10^{23}\) atoms, which gives us \(6.022 \times 10^{22}\) uranium atoms.

Molar Mass

The molar mass links moles to grams and is a fundamental concept for performing chemical calculations. It is defined as the mass of one mole of a substance and has units of grams per mole (g/mol). For any element, the molar mass in g/mol is numerically equal to its atomic mass in atomic mass units (amu), which can be found on the periodic table.

For uranium, with a molar mass of approximately \(238 \text{ g/mol}\), if we have \(0.10\) mole of uranium, we calculate the mass by multiplying the number of moles by the molar mass: \(0.10\text{ moles} \times 238\text{ g/mol}\) equaling \(23.8\) grams of uranium.

Atomic Mass Units

Atomic mass units (amu) represent the standard unit for expressing the mass of atoms and molecules. It's defined on the scale where a carbon-12 atom has an exact mass of 12 amu. Because atoms are incredibly light, this unit allows for a more convenient expression of atomic mass.

Moreover, the atomic mass unit creates an elegant link to the mole concept. The atomic mass of an element in amu is virtually equivalent to the molar mass of that element when measured in grams per mole. This means when we say an atom of uranium has an atomic mass of approximately \(238\) amu, we are also conveying that one mole of uranium atoms weighs \(238\) grams.

Chemical Calculations

Performing accurate chemical calculations is vital in chemistry. These calculations often revolve around the mole concept, requiring knowledge of Avogadro's number, the molar mass, and atomic mass units. Whether you're predicting the outcome of a reaction, determining the concentration of a solution, or simply finding how many atoms are in a given sample, these calculations enable chemists to quantify substances and predict behaviors in a controlled and measurable way.

For example, in the exercise, finding the number of atoms required the use of Avogadro's number, while the molar mass was used to convert moles to grams. These conversions are fundamental skills in chemistry that allow students and professionals alike to effectively communicate and understand quantities in the world of atoms and molecules.