Problem 31 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A compound contains 28 percent of nitrogen and 72 percent of metal by weight. 3 atoms of metal combine with 2 atoms of \(\mathrm{N}\). Find the atomic weight of metal.

Step-by-Step Solution

Answer

The atomic weight of the metal is 24.

Step 1: Assign Variables

Let the atomic weight of metal be \( M \), and the atomic weight of nitrogen be \( N = 14 \) (since this is the known atomic weight of nitrogen). We are given that there is 72% metal and 28% nitrogen in the compound by weight.

Step 2: Set Up the Ratio

According to the problem, 3 atoms of metal combine with 2 atoms of nitrogen. Hence, the mass ratio can be expressed as \( 3M \) (mass of metal) and \( 2 imes 14 = 28 \) (mass of nitrogen).

Step 3: Express the Weight Percentage in Equations

Based on the weight percentages, the equation for the metal and nitrogen ratio is given by:\[\frac{3M}{3M + 28} = 0.72\]This equation represents the metal part of the compound.

Step 4: Solve for \( M \)

Rearrange the equation from Step 3 to solve for \( M \):\[3M = 0.72(3M + 28)\]Expand and simplify:\[3M = 2.16M + 20.16\]Subtract \( 2.16M \) from both sides:\[0.84M = 20.16\]Now, solve for \( M \):\[M = \frac{20.16}{0.84} = 24\]

Step 5: Verify the Solution

Check the solution by confirming that the calculated percentage matches the original ratio:Substitute \( M = 24 \) back and calculate:\[ 3 \times 24 = 72, \quad \text{and} \quad 3M + 28 = 100 \]The percentage of metal is indeed 72%, verifying our solution is correct.

Key Concepts

Molecular Composition

Molecular composition refers to the types and numbers of atoms present in a compound. It helps to describe how the elements are grouped and combined to form a compound.
In chemistry, understanding molecular composition is crucial as it provides insight into the nature and behavior of substances. By knowing which atoms and how many of each are present, scientists can predict various properties of the compound, like its reactivity or stability.
In our exercise, we have a compound that is made up of nitrogen (\( \mathrm{N} \ \)) and metal atoms. Specifically, the compound comprises 2 nitrogen atoms for every 3 metal atoms. This ratio is integral in determining the stoichiometric calculations that follow, as it shows how the atoms in the compound are organized and combined. The notation of \( 3 \) atoms of metal and \( 2 \) atoms of nitrogen conveys not just a simple count, but the fixed proportion that these elements follow in that specific molecular structure.

Weight Percentage

Weight percentage is a way to describe the proportion of each element in a compound based on its mass contribution relative to the total mass of the compound.
It helps chemists and students alike understand the make-up of a compound in terms of weight, which is sometimes more intuitive than thinking in terms of moles or atomic counts.
In our example, the compound contains 72% metal and 28% nitrogen by weight. This means if you had 100 grams of the compound, 72 grams would be metal, and 28 grams would be nitrogen.
The given percentages are crucial for solving the problem, as they are used to set up an equation that balances the mass contributions of each element according to their atomic and molecular presence. By utilizing the weight percentage equation, \( \frac{3M}{3M + 28} = 0.72 \), one can easily calculate unknowns like the atomic weight of the metal, effectively linking the apparent percentage to tangible atomic weights.

Chemical Stoichiometry

Chemical stoichiometry involves the study of relative quantities of reactants and products in chemical reactions and compounds. It provides the quantitative relationship among substances as they undergo chemical changes.
This branch of chemistry lets us calculate necessary amounts and ratios, allowing us to predict how much of a substance is needed or will be produced in a chemical reaction.
In this problem, stoichiometry is used to relate the number of atoms and their weights. By knowing that \( 3 \) atoms of metal combine with \( 2 \) atoms of nitrogen, we set up equations that reflect these relationships.
The stoichiometry is directly applied when we write the equation \( 3M = 2.16M + 20.16 \). Solving this equation gives us the atomic weight of the metal, aligning the number of atoms with their collective weight percentages in the compound.
Understanding these connections helps elucidate how fundamental stoichiometry is in predicting and confirming the characteristics of a compound.