Question

In acting as a reducing agent a piece of metal \(\mathrm{M}\), weighing \(16.00 \mathrm{~g}\), gives up \(2.25 \times 10^{23}\) electrons. What is the weight of one equivalent of the metal?

Short answer

The weight of one equivalent of metal M can be calculated using the following steps: 1. Calculate the moles of electrons transferred: Moles of electrons = (2.25 x 10^23) / (6.022 x 10^23) 2. Find the mole-to-electron ratio using molar mass: mole-to-electron ratio = moles of M / 1 mole of electrons 3. Calculate the moles of metal M using the ratio: Moles of M = Moles of electrons * mole-to-electron ratio 4. Find the molar mass of metal M: Molar mass of M = 16.00 g / moles of M 5. Calculate the weight of one equivalent of metal M: Weight of one equivalent of M = Molar mass of M * mole-to-electron ratio

Step-by-step solution

Find moles of electrons transferred

First, we need to determine the moles of electrons that have been transferred from 16.00 g of the metal. We can use Avogadro's number (6.022 x 10^23) for this purpose. The moles of electrons can be calculated as follows: Moles of electrons = (2.25 x 10^23) / (6.022 x 10^23)

Determine mole-to-electron ratio

Next, we need to find the ratio of moles of metal M to moles of electrons. To do this, we can use the given mass of the metal (16.00 g) and the molar mass of the metal. Given that molar mass = mass / moles, we can write the equation like this: Molar mass of M = 16.00 g / moles of M We need to find the mole-to-electron ratio first by considering one mole of electrons: mole-to-electron ratio = moles of M / 1 mole of electrons

Calculate moles of metal M

To calculate the moles of metal M, we can use the ratio derived in Step 2: Moles of M = Moles of electrons (from Step 1) * mole-to-electron ratio

Find the molar mass of metal M

Now that we have the moles of metal M, we can find the molar mass of the metal using the equation: Molar mass of M = 16.00 g / moles of M (from Step 3)

Calculate the weight of one equivalent of metal M

Finally, we can calculate the weight of one equivalent of metal M by multiplying the molar mass of M (found in step 4) by the mole-to-electron ratio (from Step 2). Weight of one equivalent of M = Molar mass of M (from step 4) * mole-to-electron ratio (from Step 2)

Key concepts

Mole-to-Electron Ratio

The mole-to-electron ratio is a concept critical in stoichiometry, which is the study of the quantitative relationships between the amounts of reactants and products in a chemical reaction. Mole-to-electron ratio refers to the number of moles of a substance compared to the number of moles of electrons transferred during a redox reaction. Specifically, it is a comparison where one mole of electrons equals Avogadro's number of electrons. Understanding this ratio is important because it allows chemists to determine how much of a substance is needed to react completely with a specific number of electrons.

In our example calculation, determining the mole-to-electron ratio for metal M informs us how many moles of M react with one mole of electrons. By establishing this ratio, we unlock the relationship required for further calculations on the weight of one equivalent of the metal.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry, representing the number of units (atoms, molecules, ions, or other particles) in one mole of a substance. The value is approximately equal to 6.022 x 10^23. Avogadro's number is used to convert between the microscopic scale, involving individual atoms or molecules, and the macroscopic scale, which deals with amounts of substances that can be weighed and handled in a laboratory.

When working with redox reactions, electrons are often involved in the transfer process. Avogadro's number allows us to equate one mole of electrons to an actual quantity—essentially serving as a bridge between the unseen, incredibly small particles, and measurements that we can practically use and understand.

Molar Mass

Molar mass is the weight of one mole of a substance and is expressed in grams per mole (g/mol). It can be calculated by summing the atomic weights of all the atoms in a molecule of that substance. The molar mass is an essential concept for converting between moles and grams, which is an everyday necessity in chemistry for understanding the composition of substances and how they will react with each other.

In our exercise, we calculate the molar mass of metal M by taking the given mass of metal M and dividing it by the number of moles of M. The molar mass provides the basis for determining the equivalent weight of the metal, which is a step in understanding the substance's reactivity and how much of it will react with a given amount of another substance.

Reducing Agent

A reducing agent, in the context of a chemical reaction, is a substance that donates electrons to another species, causing itself to be oxidized. This makes the reducing agent a critical component in redox reactions where it can facilitate the reduction of another substance by losing electrons. The strength of a reducing agent is gauged by its potential to liberate electrons, which is depicted in its placement within the electrochemical series.

In the provided exercise, the piece of metal M acts as a reducing agent because it gives up electrons. By quantifying the number of electrons the metal releases, we can determine the stoichiometry of the reaction and in turn, figure out the equivalent weight of the metal. This weight tells us how much of the metal would be needed to donate a specific quantity of electrons, or in other words, how much of it is required to completely react with a corresponding oxidizing agent.