Question

Urea \(\left(\mathrm{NH}_{2} \mathrm{CONH}_{2}\right)\), an important nitrogen fertilizer, is produced industrially by the reaction $$2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{NH}_{2} \mathrm{CONH}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l)$$ Given that \(\Delta G^{\circ}=-13.6 \mathrm{~kJ}\), calculate \(\Delta G\) at \(25^{\circ} \mathrm{C}\) for the following sets of conditions: (a) \(10 \mathrm{~atm} \mathrm{NH}_{3}, 10 \mathrm{~atm} \mathrm{CO}_{2}, 1.0 \mathrm{M} \mathrm{NH}_{2} \mathrm{CONH}_{2}\) (b) \(0.10 \mathrm{~atm} \mathrm{NH}_{3}, 0.10 \mathrm{~atm} \mathrm{CO}_{2}, 1.0 \mathrm{M} \mathrm{NH}_{2} \mathrm{CONH}_{2}\) (c) Is the reaction spontaneous for the conditions in part (a) and/or part (b)?

Short answer

(a) \(\Delta G = -30.7 \mathrm{~kJ}\), spontaneous; (b) \(\Delta G = 3.5 \mathrm{~kJ}\), non-spontaneous.

Step-by-step solution

Understanding the Reaction and Given Data

We are given the reaction: \(2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{NH}_{2}\mathrm{CONH}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\) with \(\Delta G^{\circ}=-13.6 \mathrm{~kJ}\). We need to find \(\Delta G\) at \(25^{\circ} \mathrm{C}\) under different conditions. \(\Delta G\) is given by the equation \(\Delta G = \Delta G^{\circ} + RT \ln Q\), where \(R\) is the gas constant \(8.314 \mathrm{~J/mol \cdot K}\), \(T\) is the temperature in Kelvin, and \(Q\) is the reaction quotient.

Calculating Reaction Quotient (Q) for Part (a)

For part (a), \(Q\) is calculated using \(Q = \frac{[\mathrm{NH}_2\mathrm{CONH}_2]}{(P_{\mathrm{NH}_3})^2 \times P_{\mathrm{CO}_2}}\). Substituting the given concentrations and pressures: \([\mathrm{NH}_2\mathrm{CONH}_2] = 1.0 \mathrm{M}\), \(P_{\mathrm{NH}_3} = 10 \mathrm{~atm}\), and \(P_{\mathrm{CO}_2} = 10 \mathrm{~atm}\), we get \(Q = \frac{1.0}{10^2 \times 10} = 0.001\).

Calculating \(\Delta G\) for Part (a)

Using the equation \(\Delta G = \Delta G^{\circ} + RT \ln Q\) for part (a), where \(R = 8.314 \times 10^{-3} \mathrm{~kJ/mol \cdot K}\) to match the units of \(\Delta G^{\circ}\), and \(T = 298 \mathrm{~K}\) (since \(25^{\circ} \mathrm{C} = 298 \mathrm{~K}\)), we calculate:\[\Delta G = -13.6 + (8.314 \times 10^{-3} \times 298) \ln(0.001)\]Calculate \(\ln(0.001) = -6.907\), then compute:\[\Delta G = -13.6 + (2.478172 \times -6.907)\]\[\Delta G = -13.6 - 17.126 \approx -30.7 \mathrm{~kJ}\]

Calculating Reaction Quotient (Q) for Part (b)

For part (b), the reaction quotient \(Q\) is calculated similarly:\(Q = \frac{1.0}{(0.10)^2 \times 0.10}\). This gives \(Q = \frac{1.0}{0.001} = 1000\).

Calculating \(\Delta G\) for Part (b)

Using the equation \(\Delta G = \Delta G^{\circ} + RT \ln Q\) for part (b):\[\Delta G = -13.6 + (8.314 \times 10^{-3} \times 298) \ln(1000)\]Calculate \(\ln(1000) = 6.907\), then compute:\[\Delta G = -13.6 + (2.478172 \times 6.907)\]\[\Delta G = -13.6 + 17.126 \approx 3.5 \mathrm{~kJ}\]

Determining Spontaneity

A reaction is spontaneous if \(\Delta G < 0\). For part (a), \(\Delta G = -30.7 \mathrm{~kJ}\) which is negative, indicating spontaneity. For part (b), \(\Delta G = 3.5 \mathrm{~kJ}\) which is positive, indicating non-spontaneity.

Key concepts

Urea Production

Urea is an essential compound in agriculture, commonly used as a nitrogen-rich fertilizer. The production of urea occurs through an industrial process that involves a chemical reaction between ammonia \(\text{(NH}_3\text{)}\) and carbon dioxide \(\text{(CO}_2\text{)}\). This reaction produces urea \(\text{(NH}_2\text{CONH}_2)\) and water as a byproduct. Industrially, this process is conducted under high pressure and temperature to optimize yield and efficiency.

• Ammonia and carbon dioxide react in gaseous forms.
• Urea is produced in aqueous solution form, which further undergoes concentration and evaporation processes.
• This production is crucial for sustainable agriculture, providing a consistent and efficient source of nitrogen.

Understanding the reaction conditions and exploration of reaction spontaneity can further improve the industrial production of this valuable compound, aiming for maximum output with minimal energy input.

Reaction Spontaneity

Reaction spontaneity is a concept rooted in thermodynamics and is determined by Gibbs Free Energy \(\Delta G\). A reaction is spontaneous if it can proceed on its own, without outside energy input, under given conditions.
To determine spontaneity, the following criteria are used:

• If \(\Delta G < 0\), the reaction is spontaneous.
• If \(\Delta G = 0\), the reaction is at equilibrium.
• If \(\Delta G > 0\), the reaction is non-spontaneous.

In the context of urea production:

For part (a), where the pressures of reactants are relatively high, \(\Delta G\) was found to be negative (\(-30.7 \text{kJ/mol}\)), indicating that the reaction is spontaneous under these conditions.

Meanwhile, in part (b), with much lower reactant pressures, \(\Delta G\) was positive (\(3.5 \text{kJ/mol}\)), showing that the reaction is non-spontaneous. Understanding this is crucial for industrial processes to ensure the reaction proceeds efficiently, requiring specific conditions that favor spontaneity.

Chemical Equilibrium

Chemical equilibrium is an essential concept in understanding how reactions behave under different conditions. It represents a state where the forward and reverse reactions occur at the same rate, maintaining constant concentrations of reactants and products.

In terms of the urea production reaction, equilibrium can be influenced by factors such as pressure, temperature, and concentration of reactants and products.

One key factor in determining chemical equilibrium is the reaction quotient \(Q\), which is calculated by:

• Using concentrations for solutions and partial pressures for gases.
• Comparing \(Q\) to the equilibrium constant \(K\):
  • If \(Q < K\), the reaction will proceed forward to reach equilibrium.
  • If \(Q > K\), the reaction will proceed backward.

In the example given, the values of \(Q\) calculated for the different conditions directly influenced the \(\Delta G\) values obtained, thus affecting whether the reaction was spontaneous or not. Understanding chemical equilibrium allows industries to manipulate conditions to ensure optimal reaction completion while maintaining cost-effectiveness.