Question

Arrange these three samples from smallest to largest in terms of number of representative particles: \(1.25 \times 10^{25}\) atoms of zinc(Zn), 3.56 mol of iron (Fe), and \(6.78 \times 10^{22}\) molecules of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\)

Short answer

The samples arranged in ascending order based on the number of representative particles are: Glucose molecules (\(6.78 \times 10^{22}\) particles), Iron particles (\(2.142 \times 10^{24}\) particles), and Zinc atoms (\(1.25 \times 10^{25}\) particles).

Step-by-step solution

Calculate the number of representative particles in each sample.

The number of representative particles in each sample are already given for the zinc atoms and glucose molecules, which are \(1.25 \times 10^{25}\) atoms of zinc and \(6.78 \times 10^{22}\) molecules of glucose, respectively. For the iron sample, we need to convert the given moles into a number of representative particles using Avogadro's constant, \(N_A = 6.022 \times 10^{23}\ \text{particles/mol}\). Number of iron representative particles = Moles × Avogadro's constant Number of iron representative particles = 3.56 mol × \(6.022 \times 10^{23}\ \text{particles/mol}\) Number of iron representative particles = \(2.142 \times 10^{24}\) particles

Compare the number of representative particles in each sample.

Now, let's compare the number of particles for each sample. Zinc atoms: \(1.25 \times 10^{25}\) particles Iron particles: \(2.142 \times 10^{24}\) particles Glucose molecules: \(6.78 \times 10^{22}\) particles

Arrange the samples in ascending order.

By comparing the number of representative particles, we can arrange the samples in ascending order as follows: 1. Glucose molecules: \(6.78 \times 10^{22}\) particles 2. Iron particles: \(2.142 \times 10^{24}\) particles 3. Zinc atoms: \(1.25 \times 10^{25}\) particles Thus, the order will be Glucose molecules < Iron particles < Zinc atoms.

Key concepts

Avogadro's Constant

Avogadro's Constant, represented as \(N_A\), is an integral part of chemistry, especially when it comes to understanding the mole concept. It is named after the scientist Amedeo Avogadro and is defined as the number of atoms, ions, or molecules in one mole of a substance. The fixed numerical value of Avogadro's Constant is \(6.022 \times 10^{23}\) particles per mole.

This constant enables scientists and students alike to relate the macroscopic measurements of substances to the microscopic scale of atoms and molecules. For instance, if you have a sample with a mass that equals the substance's molar mass (in grams), you know that this sample contains exactly one mole, or \(6.022 \times 10^{23}\) representative particles, whether these particles are atoms, molecules, or ions.

Understanding Avogadro's Constant is crucial because it serves as a bridge between the mass of a substance and the number of particles it contains. This concept is at the heart of stoichiometry, which involves calculations based on the balanced equations in chemical reactions.

Moles to Particles Conversion

Moles to Particles Conversion is a fundamental task in chemistry, involving the translation between the amount of a substance measured in moles to the actual number of individual atoms, molecules, or ions it contains. To perform this conversion, one simply multiplies the number of moles by Avogadro's Constant (\(6.022 \times 10^{23}\) particles per mole).

The equation for this conversion is:
\[ \text{Number of particles} = \text{Number of moles} \times N_A \]
Let's use the iron sample from our exercise as an example. We're given that we have 3.56 moles of iron. The conversion would be as follows:
\[ \text{Number of iron representative particles} = 3.56 \text{ mol} \times 6.022 \times 10^{23} \text{ particles/mol} \]
After performing the multiplication, one obtains the number of iron atoms. Such calculations are crucial for determining the exact number of particles involved in reactions, preparing solutions with a precise concentration of participants, or simply comparing the quantity of substance across different materials.

Comparing Substances by Particle Number

Comparing Substances by Particle Number allows chemists to understand the relationship between different quantities of materials on a molecular level, which is necessary when discussing chemical reactions and stoichiometry.

As substances can be in various phases and consist of different types of representative particles, direct mass comparison might not be helpful. Instead, comparing the actual number of atoms, ions, or molecules provides a common ground for such comparisons. By calculating the number of representative particles using Moles to Particles Conversion, we can line up different substances and see which sample has the greater amount on the particle level.

In our exercise, after converting the number of moles of iron to particles and comparing those with the number of zinc atoms and glucose molecules, we arranged the samples by their particle count:
1. Glucose molecules: \(6.78 \times 10^{22}\) particles
2. Iron particles: \(2.142 \times 10^{24}\) particles
3. Zinc atoms: \(1.25 \times 10^{25}\) particles

This ranking provides valuable insight into the relative number of chemical entities available in each sample, which is essential in understanding concepts like reaction yield, limiting reactant, and theoretical yields in chemical reactions.