Question

In the synthesis of ammonia, if \(-\Delta\left[\mathrm{H}_{2}\right] / \Delta t=\) \(4.5 \times 10^{-4} \mathrm{mol} / \mathrm{L} \cdot \mathrm{min},\) what is \(\Delta\left[\mathrm{NH}_{3}\right] / \Delta t ?\) $$\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{g})$$

Short answer

The rate of formation of \( \mathrm{NH}_{3} \) is \( 3.0 \times 10^{-4} \mathrm{mol} / \mathrm{L} \cdot \mathrm{min} \).

Step-by-step solution

Identify the Chemical Equation and Rates

The given chemical reaction is \( \mathrm{N}_{2}(\mathrm{g}) + 3 \mathrm{H}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{g}) \). We are provided with the rate of consumption of \( \mathrm{H}_{2} \), which is \(-\Delta \left[ \mathrm{H}_{2} \right] / \Delta t = 4.5 \times 10^{-4} \mathrm{mol} / \mathrm{L} \cdot \mathrm{min} \). We need to find the rate of formation of \( \mathrm{NH}_{3} \), i.e., \( \Delta \left[ \mathrm{NH}_{3} \right] / \Delta t \).

Use Stoichiometry to Relate Rates

According to the balanced equation, 3 moles of \( \mathrm{H}_{2} \) are needed to form 2 moles of \( \mathrm{NH}_{3} \). Hence, the rate of formation of \( \mathrm{NH}_{3} \) is \( \frac{2}{3} \) times the rate of consumption of \( \mathrm{H}_{2} \).

Calculate the Rate of Formation of Ammonia

To find \( \Delta \left[ \mathrm{NH}_{3} \right] / \Delta t \), use the rate relationship: \( \Delta \left[ \mathrm{NH}_{3} \right] / \Delta t = \frac{2}{3} \left(- \Delta \left[ \mathrm{H}_{2} \right] / \Delta t\right) \). Substituting the given rate of \( \mathrm{H}_{2} \), we have: \[ \Delta \left[ \mathrm{NH}_{3} \right] / \Delta t = \frac{2}{3} \times 4.5 \times 10^{-4} \mathrm{mol} / \mathrm{L} \cdot \mathrm{min} \]Calculating gives: \[ \Delta \left[ \mathrm{NH}_{3} \right] / \Delta t = 3.0 \times 10^{-4} \mathrm{mol} / \mathrm{L} \cdot \mathrm{min} \]

Verify the Calculation

Double check the calculation by multiplying \( \frac{2}{3} \) with \( 4.5 \times 10^{-4} \), which should result in \( 3.0 \times 10^{-4} \). This confirms that the rate of formation of \( \mathrm{NH}_{3} \) is correct.

Key concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that involves the quantitative relationship between reactants and products in a chemical reaction. It allows us to predict how much of a substance is needed or produced. In the context of the ammonia synthesis reaction, stoichiometry tells us that for every 1 mole of nitrogen (_2), 3 moles of hydrogen (_2) are required to produce 2 moles of ammonia (H_3).
This relationship is crucial because it helps us understand the proportion of reactants needed and the amount of product formed. By using stoichiometry, chemists can calculate the rates of reactions and the quantities involved. In our worked example, we used the balanced chemical equation to determine that the rate of ammonia formation is \(\frac{2}{3}\) times the rate at which hydrogen gas is consumed.
To apply stoichiometry in reaction rates, we rely on conversion factors derived from the coefficients in a balanced chemical equation. Here, the conversion factor comes from the numbers in front of the substances: 3 for hydrogen and 2 for ammonia. Thus, stoichiometry serves as a bridge to connect the macroscopic predictions we can make about reactions with the microscopic world of atoms and molecules.

Chemical Equations

A chemical equation is a symbolic representation of a chemical reaction. It provides information about which reactants are involved and what products are formed. In our example, the balanced chemical equation is \[ \mathrm{N}_2(\mathrm{g}) + 3 \mathrm{H}_2(\mathrm{g}) \rightarrow 2 \mathrm{NH}_3(\mathrm{g}) \].
The equation tells us that one molecule of nitrogen reacts with three molecules of hydrogen to form two molecules of ammonia. The state symbols (\(\mathrm{g}\)) indicate that all substances involved are gases. Balancing chemical equations ensures that the same number of each type of atom appears on both sides of the equation, adhering to the law of conservation of mass.
To read a chemical equation effectively, you need to identify the reactants (substances that start the reaction) and products (the substances formed). Additionally, the coefficients (numbers in front of chemical formulas) tell us the ratio in which substances react or are produced. In our exercise, understanding these equations is key to solving the problem because the coefficients directly inform the stoichiometric relationships and reaction rates.

Ammonia Synthesis

Ammonia synthesis, commonly known as the Haber-Bosch process, is an industrial-scale method to produce ammonia from nitrogen and hydrogen. The general reaction is:\( \mathrm{N}_2(\mathrm{g}) + 3 \mathrm{H}_2(\mathrm{g}) \rightarrow 2 \mathrm{NH}_3(\mathrm{g}) \).
This reaction is of immense importance because ammonia is a critical component in fertilizers, thereby playing a vital role in agriculture and food production worldwide. The process was developed by Fritz Haber and Carl Bosch in the early 20th century, revolutionizing chemical manufacturing and helping to meet the world's food demands.
The synthesis of ammonia is carried out at high pressures and temperatures, typically using an iron catalyst to speed up the reaction. The balanced chemical equation we've examined demonstrates the conversion of nitrogen and hydrogen into ammonia. Each molecule of nitrogen reacts with three molecules of hydrogen, illustrating an elegant example of efficient chemical transformation, underscoring the significance of stoichiometry and balanced equations.
Ammonia synthesis not only underscores the importance of chemical equations in predicting reaction outcomes but also highlights the practical applications of stoichiometry in large-scale chemical processes.