Question

Urea \(\left(\mathrm{H}_{2} \mathrm{NCONH}_{2}\right)\) is used extensively as a nitrogen source in fertilizers. It is produced commercially from the reaction of ammonia and carbon dioxide: Ammonia gas at \(223^{\circ} \mathrm{C}\) and 90 . atm flows into a reactor at a rate of \(500 . \mathrm{L} / \mathrm{min}\). Carbon dioxide at \(223^{\circ} \mathrm{C}\) and 45 atm flows into the reactor at a rate of \(600 . \mathrm{L} / \mathrm{min}\). What mass of urea is produced per minute by this reaction assuming \(100 \%\) yield?

Short answer

The mass of urea produced per minute can be calculated by first writing the balanced chemical equation: \(2 \mathrm{NH}_{3}(\mathrm{g}) + \mathrm{CO}_{2}(\mathrm{g}) \rightarrow \mathrm{H}_{2}\mathrm{NCONH}_{2}(\mathrm{s})+ \mathrm{H}_{2}\mathrm{O}(\mathrm{g})\). Then, convert the flow rates of ammonia and carbon dioxide to moles per minute using the ideal gas law. Identify the limiting reactant and use the stoichiometry of the reaction to determine the moles of urea produced. Finally, convert the moles of urea to mass using its molar mass (60.06 g/mol).

Step-by-step solution

Write the balanced chemical equation

The balanced equation for the reaction can be written as: \[2 \mathrm{NH}_{3}(\mathrm{g}) + \mathrm{CO}_{2}(\mathrm{g}) \rightarrow \mathrm{H}_{2}\mathrm{NCONH}_{2}(\mathrm{s})+ \mathrm{H}_{2}\mathrm{O}(\mathrm{g})\]

Convert flow rates to moles

Using the ideal gas law, we can convert the flow rates of ammonia and carbon dioxide to moles per minute: \[PV = nRT\] \[n = \frac{PV}{RT}\] For ammonia: Pressure, \(P_{NH_{3}}\) = 90 atm Volume, \(V_{NH_{3}}\) = 500 L/min Temperature, \(T_{NH_{3}}\) = 223°C + 273 = 496 K \(R\) = 0.0821 L.atm/mol.K (Ideal Gas Constant) \[n_{NH_{3}} = \frac{P_{NH_{3}}V_{NH_{3}}}{RT_{NH_{3}}}\] \[n_{NH_{3}} = \frac{90 \cdot 500}{0.0821 \cdot 496}\] For carbon dioxide: Pressure, \(P_{CO_{2}}\) = 45 atm Volume, \(V_{CO_{2}}\) = 600 L/min Temperature, \(T_{CO_{2}}\) = 223°C + 273 = 496 K \[n_{CO_{2}} = \frac{P_{CO_{2}}V_{CO_{2}}}{RT_{CO_{2}}}\] \[n_{CO_{2}} = \frac{45 \cdot 600}{0.0821 \cdot 496}\]

Identify the limiting reactant

Using the stoichiometry of the balanced chemical equation, we can determine which reactant is the limiting reactant by comparing the mole ratios. For every 2 moles of NH3, 1 mole of CO2 is reacted. To find the limiting reactant, divide the moles per minute for each reactant by the stoichiometric ratio: \[\frac{n_{NH_{3}}}{2}\] and \[\frac{n_{CO_{2}}}{1}\] Compare these values. The smaller one indicates the limiting reactant.

Calculate the moles of urea produced

Using the stoichiometry of the balanced chemical equation, find the moles of urea produced per minute based on the moles of the limiting reactant. If NH3 is the limiting reactant, calculate as follows: \[moles\:of\:urea = \frac{1}{2} \cdot n_{NH_{3}}\] If CO2 is the limiting reactant, calculate as follows: \[moles\:of\:urea = 1 \cdot n_{CO_{2}}\]

Convert moles of urea to mass

Now, by knowing the moles of urea produced per minute, convert it into mass using the molar mass of urea. Molar mass of urea = 60.06 g/mol \[mass\:of\:urea\:produced = moles\:of\:urea \cdot 60.06\:g/mol\] This will provide the mass of urea produced per minute, assuming a 100% yield.

Key concepts

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry involves the mathematical relationship between the quantities of reactants and products in a chemical reaction. It's based on the principle of the conservation of mass, which states that matter cannot be created or destroyed in a chemical reaction. Stoichiometry calculations require a balanced chemical equation, providing the basis for determining how much of each reactant is needed to form a desired amount of product.

In the case of urea synthesis, the balanced equation is pivotal, as it informs us that two moles of ammonia (NH_{3}) react with one mole of carbon dioxide (CO_2) to produce one mole of urea (H_2NCONH_2) and one mole of water (H_2O). This stoichiometric ratio is applied when identifying the limiting reactant and calculating the amount of urea that can be produced from given quantities of reactants. Understanding this component of the reaction is essential to predict yields and to scale up production for industrial purposes.

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the behavior of an ideal gas. Expressed as PV = nRT, where P is pressure, V is volume, n is the amount of substance in moles, R is the ideal gas constant, and T is the temperature in Kelvin, it enables chemists to relate the physical properties of a gaseous reactant or product. In our urea synthesis example, the ideal gas law is used to convert the given flow rates of ammonia and carbon dioxide, measured in liters per minute at specific temperatures and pressures, into moles per minute.

This step is essential for determining the stoichiometry of the reactants' flows into the reactor. Since gaseous substances are often manipulated under different conditions in industrial processes, the ability to convert between units using the ideal gas law is valuable for ensuring accurate stoichiometric calculations in reactive systems.

Limiting Reactant Determination

The concept of the limiting reactant is key in chemical reactions and is crucial in calculating the theoretical yield of the product. The limiting reactant is the reactant that is consumed first, thus preventing more product from forming once it has been used up. Determining the limiting reactant involves comparing the amount of each reactant that is available to the stoichiometric requirements of the reaction.

In our urea synthesis problem, after converting the flow rates of ammonia and carbon dioxide to moles, we need to compare these values in the context of our balanced equation. As the chemical equation dictates that two ammonia molecules react with a single carbon dioxide molecule, the comparison of mole ratios will reveal which reactant is limiting. This determination is the bottleneck of the synthesis process, defining the maximum amount of urea that could be produced and is a critical step in optimizing the efficiency of industrial chemical processes.

Molar Mass Calculation

To convert moles of a substance to mass, one needs to know the substance's molar mass, which is the mass of one mole of that substance. The molar mass is calculated by summing the masses of all the atoms in a molecule as given by the periodic table, typically expressed in grams per mole (g/mol).

For urea (H_2NCONH_2), the molar mass calculation takes into account the atomic masses of nitrogen, carbon, oxygen, and hydrogen. The molar mass is used to convert the moles of urea, computed based on the limiting reactant, into grams. This conversion is the final step in our problem, enabling us to answer the practical question of how much urea mass is produced per minute. Mastery of molar mass calculations is thus indispensable for converting between different units in chemistry, ensuring accurate handling and measurement of substances for both academic and practical applications.