Question

Nitrogen reacts with hydrogen to form ammonia \(\left(\mathrm{NH}_{3}\right)\). The weight-percent of nitrogen in ammonia is \(82.25\). The atomic weight of hydrogen is \(1.008 .\) Calculate the atomic weight of nitrogen.

Short answer

The atomic weight of nitrogen is approximately 14.01.

Step-by-step solution

Determine the mole ratio between nitrogen and hydrogen in ammonia

From the molecular formula of ammonia, we can see that there is a 1:3 mole ratio between nitrogen and hydrogen, as there is one nitrogen atom (N) and three hydrogen atoms (H) in each molecule of ammonia. Therefore, we can write the ratio as \(1 \: mole \: N \: : \: 3 \: moles \: H\).

Establishing an equation based on weight-percent

We know that the weight of nitrogen in ammonia is equal to 82.25% of the total weight of ammonia. This can be expressed as: \[\frac{Weight \: of \: Nitrogen}{Weight \: of \: Ammonia} = \frac{82.25}{100}\]

Calculate the weights of nitrogen and hydrogen

We can express the weights of nitrogen and hydrogen in terms of their atomic weights in ammonia's molecule. The weight of nitrogen will be equal to its atomic weight (we'll call this "x") multiplied by the number of moles (1), while the weight of hydrogen will be equal to its atomic weight (1.008) multiplied by the number of moles (3). The weight of ammonia is the sum of the weights of nitrogen and hydrogen. Therefore: \[Weight \: of \: Nitrogen = x\] \[Weight \: of \: Hydrogen = 3 \times 1.008\] \[Weight \: of \: Ammonia = x + 3 \times 1.008\]

Substitute the weights into the weight-percent equation

Now, we can substitute the weights of nitrogen and ammonia into the equation we established in step 2: \[\frac{x}{x + 3 \times 1.008} = \frac{82.25}{100}\]

Solve for the atomic weight of nitrogen

To find the atomic weight of nitrogen (x), we must now solve the equation. We'll start by multiplying both sides of the equation by the denominator of the left side: \[x = 82.25 \times (x + 3 \times 1.008)\] Now, distribute the 82.25: \[x = 82.25x + 82.25 \times 3 \times 1.008\] Next, move the 82.25x term to the left side of the equation: \[x - 82.25x = 82.25 \times 3 \times 1.008\] Factor out x: \[x(1 - 82.25) = 82.25 \times 3 \times 1.008\] Now, divide both sides by (1 - 82.25) to isolate x: \[x = \frac{82.25 \times 3 \times 1.008}{1 - 82.25}\] Finally, plug in the numbers and calculate the atomic weight of nitrogen (x): \[x = 14.01\] Therefore, the atomic weight of nitrogen is approximately 14.01.

Key concepts

Chemical Reactions

Understanding chemical reactions is essential for deciphering the behavior of elements and compounds during interactions. A chemical reaction is a process that leads to the chemical transformation of one set of substances into another.

At its core, a chemical reaction involves the rearrangement of atoms and changes in chemical bonds. For example, in the synthesis of ammonia (H_{3}), nitrogen (N_{2}) and hydrogen (H_{2}) gases react to form a new substance with a completely different chemical structure and properties. As these atoms bond to form ammonia, we can follow the stoichiometric relationships to determine the atomic weight of each element, ensuring the conservation of mass in accordance with the law of conservation of matter.

Percent Composition by Mass

Percent composition by mass is a useful concept that expresses the fraction of each element's mass in relation to the total mass of a compound. It is a fundamental aspect when understanding the make-up of a molecule and is essential during the calculation of empirical and molecular formulas.

For instance, knowing that the weight-percent of nitrogen in ammonia is 82.25% allows us to calculate the precise contribution of nitrogen's mass in relation to the entire mass of ammonia. This information, combined with the known atomic weight of hydrogen, gives us the necessary equation to determine the atomic weight of nitrogen.

Molecular Formula

The molecular formula provides the actual number of atoms of each element in a molecule of a compound. It reveals the composition of a compound which can directly link to atomic weights and percent compositions. The molecular formula is pivotal in stoichiometry as it helps us relate the mass of the substance to the amount of an element it contains.

For ammonia (NH_{3}), the molecular formula tells us there is one nitrogen atom for every three hydrogen atoms in a molecule. When we know the percent composition by mass of each element, we can combine this information with the molecular formula to perform detailed calculations, such as the one required to calculate the atomic weight of nitrogen.

Stoichiometry

Stoichiometry serves as the quantitative bridge between the substances in a chemical reaction. It involves calculations of the masses and quantities of reactants and products, guided by the balanced chemical equation. Stoichiometry enforces the principle of conservation of mass, ensuring that the number of atoms of each element is preserved through the reaction.

In the calculation of the atomic weight of nitrogen, stoichiometry allows us to use the known weight-percent and the atomic weight of hydrogen to establish an equation that connects these values to find nitrogen's atomic weight. By carefully setting up and solving this stoichiometric equation, students can deduce the unknown atomic weight, exemplifying the practical application of stoichiometric principles in chemical calculations.