Question

A 12.0 -g sample of an element contains \(5.94 \times 10^{22}\) atoms. What is the unknown element? (Chapter 10\()\)

Short answer

To find the unknown element, follow these steps: 1. Calculate the number of moles of the given sample: \[ \text{Number of moles} = \frac{5.94 \times 10^{22} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}} \] 2. Determine the molar mass of the unknown element: \[ \text{Molar mass} = \frac{12.0 \text{ g}}{\frac{5.94 \times 10^{22} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}}} \] 3. Identify the element from the periodic table based on the molar mass.

Step-by-step solution

Calculate the number of moles of the given sample

We know that the number of atoms in the sample is \(5.94 \times 10^{22}\) atoms, which we can now convert into moles by dividing the number of atoms by Avogadro's number. \[ \text{Number of moles} = \frac{\text{Number of atoms}}{\text{Avogadro's number}} \] Now, we will use the provided data to determine the number of moles: \[ \text{Number of moles} = \frac{5.94 \times 10^{22} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}} \]

Determine the molar mass of the unknown element

Next, we will calculate the molar mass of the unknown element. To do this, we will divide the mass of the sample by the number of moles. \[ \text{Molar mass} = \frac{\text{Mass}}{\text{Number of moles}} \] Plugging in the given mass and the calculated moles, we get: \[ \text{Molar mass} = \frac{12.0 \text{ g}}{\frac{5.94 \times 10^{22} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}}} \]

Identify the element from the periodic table based on the molar mass

Now that we have calculated the molar mass of the unknown element, we can refer to the periodic table and look for an element with a similar molar mass. Once we find the element, we can correctly identify the unknown element.

Key concepts

Avogadro's Number

When dealing with atoms and molecules in chemistry, we often come across an extremely large number: Avogadro's number. It is given the symbol 'N_A' and has a value of approximately 6.022 x 10²³. Named after the Italian scientist Amedeo Avogadro, this is the number of atoms or molecules in one mole of a substance.

Why is Avogadro's number so important? It allows chemists to bridge the gap between the microscopic world (atoms and molecules) and the macroscopic world (grams and liters) that we can measure in a lab. For example, if we have a 12.0 g sample of an element that contains 5.94 x 10²² atoms, we can calculate how many moles this represents by dividing the number of atoms by Avogadro's number \(\frac{5.94 \times 10^{22}}{6.022 \times 10^{23}} \) which simplifies our understanding of quantities in chemical reactions.

Molar Mass Calculation

Molar mass serves as a bridge between the mass of a substance and the number of moles. It is defined as the mass of one mole of a given substance, usually expressed in grams per mole (g/mol). Calculating the molar mass is crucial for stoichiometric calculations in chemistry, where one needs to relate masses of different substances involved in a chemical reaction.

So, how do you calculate the molar mass from a given sample? Simply divide the mass of your sample by the number of moles in that sample. According to our exercise, the molar mass \( M \) can be calculated as follows: \( M = \frac{12.0 \text{ g}}{\frac{5.94 \times 10^{22} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}}} \). This will give the mass of one mole of the atoms, and then we can use the periodic table to identify the element with a matching molar mass.

Periodic Table Element Identification

The periodic table is a chemist's road map, showcasing all known elements organized by their atomic number, electron configuration, and recurring chemical properties. Each element is listed with its symbol and atomic mass, which is essentially the molar mass that averages out for all naturally occurring isotopes of that element.

In our exercise, after calculating the molar mass, we identify the element by matching the calculated molar mass with the atomic mass listed on the periodic table. It's a bit like a detective's work: match the 'fingerprint' (molar mass) of the unknown sample to the 'records' (the periodic table) to find the element's identity. To improve understanding, one might consider integrating visual aids, like extracting and presenting the small section of the periodic table showing where the element lies in context with its neighbors, to allow students to more closely associate the abstract concept of molar mass with the tangible layout of the periodic table.