Question

Ammonium cyanate, NH_NCO, rearranges in water to give urea, (NH\(_{2}\)) \(_{2}\) CO: $$\mathrm{NH}_{4} \mathrm{NCO}(\mathrm{aq}) \longrightarrow\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}(\mathrm{aq})$$ The rate equation for this process is "Rate \(=k\) \(\left[\mathrm{NH}_{4} \mathrm{NCO}\right]^{2}, "\) where \(k=0.0113 \mathrm{L} / \mathrm{mol} \cdot\) min. If the original concentration of \(\mathrm{NH}_{4} \mathrm{NCO}\) in solution is \(0.229 \mathrm{mol} / \mathrm{L}\) how long will it take for the concentration to decrease to \(0.180 \mathrm{mol} / \mathrm{L} ?\)

Short answer

It takes approximately 105 minutes.

Step-by-step solution

Understand the Rate Equation

The reaction given has a rate equation: \( \text{Rate} = k \left[ \mathrm{NH}_{4} \mathrm{NCO} \right]^2 \). This means the reaction follows a second-order kinetics, specifically depending on the square of the concentration of \( \mathrm{NH}_{4} \mathrm{NCO} \).

Write the Integrated Rate Law for Second-Order Reactions

For second-order reactions, the integrated rate equation is: \[ \frac{1}{[A]} = \frac{1}{[A]_0} + kt \] where \( [A] \) is the concentration at time \( t \), \( [A]_0 \) is the initial concentration, and \( k \) is the rate constant.

Substitute Known Values into the Integrated Rate Law

Given: \( [A]_0 = 0.229 \, \mathrm{mol/L} \), \( [A] = 0.180 \, \mathrm{mol/L} \), and \( k = 0.0113 \, \mathrm{L} / \mathrm{mol} \cdot \mathrm{min} \). Substitute these into the integrated rate law: \[ \frac{1}{0.180} = \frac{1}{0.229} + 0.0113t \]

Solve for Time \( t \)

Rearrange the equation to solve for \( t \): \[ t = \frac{1}{0.0113} \left( \frac{1}{0.180} - \frac{1}{0.229} \right) \]. Calculate as follows:\[ \frac{1}{0.180} = 5.555 \]\[ \frac{1}{0.229} = 4.367 \]So, \[ t = \frac{1}{0.0113} (5.555 - 4.367) \]\[ t = \frac{1}{0.0113} (1.188) \]\[ t = 105.133 \] approximately.

Key concepts

Ammonium cyanate

Ammonium cyanate, often represented as NH extsubscript{4}NCO, plays a fascinating role in chemistry, primarily because of its ability to undergo a rearrangement into urea when dissolved in water. It is an inorganic compound that illustrates the concept of chemical conversion through molecular rearrangement. This reaction is a classic example showcasing how a seemingly simple compound can transform into another with entirely different chemical properties and biological significance.

In this reaction, ammonium cyanate serves as a precursor to urea, a crucial organic compound identified by the formula \((NH_2)_2CO\). The process itself is important not just in educational settings but also in biological contexts, as it was historically significant in vitality theories during its discovery in the 19th century. This rearrangement emphasizes the unique abilities of chemical substances to change form while participating in fundamental chemical reactions.

Urea formation

Urea formation is an intriguing and significant chemical process with biological relevance. The reaction from ammonium cyanate (NH extsubscript{4}NCO) to urea ((NH extsubscript{2}) extsubscript{2}CO) is a splendid demonstration of the simple molecules achieving biological complexity.

Historically, this process was pivotal in organic chemistry, marking the discovery that organic compounds could be synthesized from inorganic materials. In 1828, Friedrich Wöhler discovered this transformation, which challenged the idea that organic compounds required a living origin, laying the foundation for organic synthesis.

• Urea is a significant player in the metabolism of nitrogen-containing substances in animals.
• It serves as a non-toxic compound used to excrete excess nitrogen primarily in the urine.
• The ability to synthesize urea artificially has profound implications in agriculture, where it is used as a fertilizer, and in medical applications as a non-toxic byproduct.

Integrated rate law

The integrated rate law is a critical tool in understanding reaction kinetics, especially for second-order reactions such as the conversion of ammonium cyanate to urea. Kinetics allows chemists to describe how the concentration of reactants decreases over time by providing a mathematical relationship.

For second-order reactions, the integrated rate law is:\[ \frac{1}{[A]} = \frac{1}{[A]_0} + kt \]where:

• \([A]\) is the concentration of the reactant at time \(t\).
• \([A]_0\) is the initial concentration.
• \(k\) is the rate constant, which reflects how quickly reactants convert to products.
• \(t\) is the time period over which the reaction occurs.

This formula reveals how the concentration of ammonium cyanate decreases quadratically over time, being directly proportional to the rate constant and time. In practical applications, calculations involving the integrated rate law help determine how long it will take for a reactant's concentration to reach a desired level, as demonstrated in solving problems with known initial and final concentrations.