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Chapter 2: Problem 14

Calculate the number of atoms in 16 grams of \(\mathrm{O}_{2}, 31\) grams of \(\mathrm{P}_{4}\), and 32 grams of \(\mathrm{S}_{2}\).

Short Answer

Expert verified
Each element contains \(6.022 \times 10^{23}\) atoms.

Step by step solution

01

Understand the Problem

We need to calculate the number of atoms in given masses of molecules: \(\mathrm{O}_2\), \(\mathrm{P}_4\), and \(\mathrm{S}_2\). This involves converting grams into moles and then using Avogadro's number to find the number of atoms.
02

Calculate Molar Mass of Each Molecule

Determine the molar mass of \(\mathrm{O}_2\), \(\mathrm{P}_4\), and \(\mathrm{S}_2\).- Molar mass of \(\mathrm{O}_2\) is \(2 \times 16 = 32 \ \text{g/mol}\).- Molar mass of \(\mathrm{P}_4\) is \(4 \times 31 = 124 \ \text{g/mol}\).- Molar mass of \(\mathrm{S}_2\) is \(2 \times 32 = 64 \ \text{g/mol}\).
03

Convert Grams to Moles

Use the molar mass to convert grams into moles for each substance.- For \(\mathrm{O}_2\), \(\frac{16}{32} = 0.5\ \text{moles}\).- For \(\mathrm{P}_4\), \(\frac{31}{124} = 0.25\ \text{moles}\).- For \(\mathrm{S}_2\), \(\frac{32}{64} = 0.5\ \text{moles}\).
04

Calculate Number of Molecules

Use Avogadro's number \(6.022 \times 10^{23}\) to find the number of molecules from the moles calculated.- \(\mathrm{O}_2\): \(0.5 \times 6.022 \times 10^{23} = 3.011 \times 10^{23}\ \text{molecules}\).- \(\mathrm{P}_4\): \(0.25 \times 6.022 \times 10^{23} = 1.5055 \times 10^{23}\ \text{molecules}\).- \(\mathrm{S}_2\): \(0.5 \times 6.022 \times 10^{23} = 3.011 \times 10^{23}\ \text{molecules}\).
05

Calculate Number of Atoms

Finally, use the molecular formula to calculate the number of atoms.- \(\mathrm{O}_2\): \(3.011 \times 10^{23}\) molecules \(\times 2 = 6.022 \times 10^{23}\ \text{atoms}\).- \(\mathrm{P}_4\): \(1.5055 \times 10^{23}\) molecules \(\times 4 = 6.022 \times 10^{23}\ \text{atoms}\).- \(\mathrm{S}_2\): \(3.011 \times 10^{23}\) molecules \(\times 2 = 6.022 \times 10^{23}\ \text{atoms}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Mass
Molar mass is a crucial concept in chemistry that helps us determine the amount of a substance. It is defined as the mass of one mole of a substance expressed in grams per mole (g/mol). To find the molar mass, you add up the atomic masses of each element in a molecule based on the periodic table.
For example, in the molecule of oxygen
  • Oxygen ( \( \text{O}_2 \)) consists of two oxygen atoms. Since the atomic mass of oxygen is approximately 16 g/mol, the molar mass of \( \text{O}_2 \) is calculated as \( 2 \times 16 = 32 \, \text{g/mol} \).
  • For phosphorus ( \( \text{P}_4 \)), the atomic mass is 31 g/mol, so the molar mass for \( \text{P}_4 \) is \( 4 \times 31 = 124 \, \text{g/mol} \).
  • Finally, sulfur ( \( \text{S}_2 \)), with an atomic mass also around 32 g/mol, results in a molar mass of \( 2 \times 32 = 64 \, \text{g/mol} \).
Knowing the molar mass allows us to convert between grams, the mass we can measure, and moles, the chemical quantity necessary for stoichiometric calculations.
Conversion of grams to moles
To perform chemical calculations, converting grams to moles is often essential. Moles are a unit for counting atoms, molecules, or ions, effectively reducing large numbers to a manageable size using Avogadro's number, which is \( 6.022 \times 10^{23} \). Here's how you do it:
You divide the given mass of your substance by its molar mass.
  • For \( \text{O}_2 \), we have 16 grams. Using its molar mass of 32 g/mol, you would calculate the moles as \( \frac{16}{32} = 0.5 \) moles.
  • For \( \text{P}_4 \), with a mass of 31 grams and a molar mass of 124 g/mol, the conversion is \( \frac{31}{124} = 0.25 \) moles.
  • For \( \text{S}_2 \), 32 grams converts into \( \frac{32}{64} = 0.5 \) moles.
This step is crucial because from here, you can use the number of moles to find the number of molecules or atoms.
Molecular Formula calculations
Once you've determined the number of moles, you can calculate the number of atoms within a sample by using the molecular formula. The molecular formula tells you how many atoms of each element are in a molecule of a compound. This is done in a few steps:
First, calculate the number of molecules by multiplying moles by Avogadro's number ( \( 6.022 \times 10^{23} \)).
  • For \( \text{O}_2 \), with 0.5 moles, the number of molecules is \( 0.5 \times 6.022 \times 10^{23} = 3.011 \times 10^{23} \) molecules.
  • \( \text{P}_4 \) with 0.25 moles yields \( 0.25 \times 6.022 \times 10^{23} = 1.5055 \times 10^{23} \) molecules.
  • \( \text{S}_2 \) also results in \( 3.011 \times 10^{23} \) molecules for 0.5 moles.
Next, use the molecular formula to transform these into the number of atoms:
  • For \( \text{O}_2 \), multiply the number of molecules by 2 (since each \( \text{O}_2 \) contains 2 oxygen atoms) to get \( 6.022 \times 10^{23} \) atoms.
  • \( \text{P}_4 \) yields the same because there are four phosphorus atoms in each molecule: \( 1.5055 \times 10^{23} \times 4 = 6.022 \times 10^{23} \) atoms.
  • Finally, \( \text{S}_2 \) follows the same pattern as oxygen, leading to \( 6.022 \times 10^{23} \) atoms.
This is how the molecular formula can be used in coordination with moles to find the actual count of atoms in a given mass of a compound.

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