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Chapter 5: Problem 77

Urea, \(\mathrm{NH}_{2} \mathrm{CONH}_{2},\) is a nitrogen fertilizer that is manufactured from ammonia and carbon dioxide. $$ 2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{NH}_{2} \mathrm{CONH}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l) $$ What volume of ammonia at \(25^{\circ} \mathrm{C}\) and \(3.00 \mathrm{~atm}\) is needed to produce \(908 \mathrm{~g}\) ( \(2 \mathrm{lb}\) ) of urea?

Short Answer

Expert verified
246.5 L of ammonia is needed.

Step by step solution

01

Determine Moles of Urea

First, find the molar mass of urea, \( \mathrm{NH}_2 \mathrm{CONH}_2 \). The molar mass is the sum of the atomic masses of all atoms in one molecule: \( 2 \times 14.01 + 4 \times 1.01 + 12.01 + 16.00 = 60.06 \text{ g/mol} \). To find the number of moles of urea produced, use the formula \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). Plug in the values: \( \frac{908 \text{ g}}{60.06 \text{ g/mol}} \approx 15.12 \text{ moles} \).
02

Determine Moles of Ammonia Required

According to the balanced chemical equation, 2 moles of \( \mathrm{NH}_3 \) are required to produce 1 mole of urea. Therefore, the moles of \( \mathrm{NH}_3 \) needed are \( 2 \times 15.12 \approx 30.24 \) moles.
03

Apply the Ideal Gas Law

Use the Ideal Gas Law to find the volume of \( \mathrm{NH}_3 \), where the formula is \( PV = nRT \). Given: \( P = 3.00 \text{ atm}, n = 30.24 \text{ moles}, T = 298 \text{ K} \) (which is \( 25^{\circ}C + 273 = 298 \text{ K} \)), and \( R = 0.0821 \text{ L atm/mol K} \). Rearrange to find \( V = \frac{nRT}{P} \).
04

Calculate the Volume of Ammonia

Insert the known values into the equation from Step 3: \( V = \frac{30.24 \times 0.0821 \times 298}{3.00} \). Calculate: \( V \approx 246.5 \text{ L} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Mass Calculation
Understanding molar mass calculation is crucial for solving chemical problems. The molar mass of a compound is the sum of the atomic masses of all the atoms in its molecular formula. For example, urea, indicated as \( \mathrm{NH}_2 \mathrm{CONH}_2 \), consists of nitrogen, carbon, oxygen, and hydrogen atoms. To calculate its molar mass, we add the molar masses of these elements:
  • Nitrogen (\( \mathrm{N} \)): 2 atoms 脳 14.01 g/mol = 28.02 g/mol
  • Hydrogen (\( \mathrm{H} \)): 4 atoms 脳 1.01 g/mol = 4.04 g/mol
  • Carbon (\( \mathrm{C} \)): 1 atom 脳 12.01 g/mol = 12.01 g/mol
  • Oxygen (\( \mathrm{O} \)): 1 atom 脳 16.00 g/mol = 16.00 g/mol
This yields a total molar mass of 60.06 g/mol. With this, you can convert a given mass of a substance into moles using the formula: \[\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\]This is essential for progressing in chemical calculations.
Chemical Reactions
Chemical reactions involve the transformation of reactants into products. The reaction we focus on here is the production of urea from ammonia and carbon dioxide:\[2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \rightarrow \mathrm{NH}_{2} \mathrm{CONH}_{2}(aq)+\mathrm{H}_{2} \mathrm{O}(l)\]In this equation, the reactants \( \mathrm{NH}_3 \) and \( \mathrm{CO}_2 \) undergo a reaction to produce urea \( \mathrm{NH}_2 \mathrm{CONH}_2 \) and water \( \mathrm{H}_2 \mathrm{O} \). Understanding chemical reactions allows us to determine how much reactant is needed or how much product we can obtain from a reaction, a process explored through stoichiometry.
Stoichiometry
Stoichiometry is the heart of chemistry that explains the quantitative relationships between reactants and products in a chemical reaction. By interpreting a balanced chemical equation, like the one for urea production, you can determine the proportion of each reactant needed or the amount of product formed.In this example, the reaction equation tells us that 2 moles of \( \mathrm{NH}_3 \) react with 1 mole of \( \mathrm{CO}_2 \) to produce 1 mole of urea \( \mathrm{NH}_2 \mathrm{CONH}_2 \). Thus, if you need to make 15.12 moles of urea, you would require:
  • 2 脳 15.12 moles = 30.24 moles of \( \mathrm{NH}_3 \)
Understanding stoichiometry is critical for predicting the amounts of substances consumed and produced in reactions.
Gas Laws
Gas laws describe the behavior of gases and are essential for understanding reactions involving gaseous substances like ammonia \( \mathrm{NH}_3 \). One fundamental gas law is the Ideal Gas Law, expressed as:\[\]\[PV = nRT\]\[\]Here:
  • \( P \) is the pressure of the gas
  • \( V \) is its volume
  • \( n \) represents the number of moles
  • \( R \) is the gas constant (0.0821 \( \text{L atm/mol K} \))
  • \( T \) is the temperature in Kelvin
By rearranging the equation to solve for \( V \) (volume), you can determine the volume of gas needed. In our ammonia example, if you have 30.24 moles at a temperature of 298 K and a pressure of 3.00 atm, you can calculate the volume needed for the reaction.
Ammonia Production
Ammonia \( \mathrm{NH}_3 \) is a key component in the production of fertilizers like urea. It is often produced through the Haber process, which combines nitrogen and hydrogen gases under high pressure and temperature in the presence of a catalyst.In industrial settings, ammonia is then reacted with carbon dioxide to produce urea. This is a crucial reaction in agriculture since urea provides a rich nitrogen source for plants. Efficient production and utilization of ammonia are both economically and environmentally significant, ensuring that crops have enough nutrients while minimizing waste and emissions.

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Most popular questions from this chapter

A mole of gas at \(0^{\circ} \mathrm{C}\) and \(760 \mathrm{mmHg}\) occupies \(22.41 \mathrm{~L}\). What is the volume at \(20^{\circ} \mathrm{C}\) and \(760 \mathrm{mmHg}\) ?

A cylinder of oxygen gas contains \(91.3 \mathrm{~g} \mathrm{O}_{2}\). If the volume of the cylinder is \(8.58 \mathrm{~L}\), what is the pressure of the \(\mathrm{O}_{2}\) if the gas temperature is \(21^{\circ} \mathrm{C} ?\)

A rigid 1.0 - \(\mathrm{L}\) container at \(75^{\circ} \mathrm{C}\) is fitted with a gas pressure gauge. A 1.0 -mol sample of ideal gas is introduced into the container. What would the pressure gauge in the container be reading in \(\mathrm{mmHg}\) ? Describe the interactions in the container that are causing the pressure. c Say the temperature in the container were increased to \(150^{\circ} \mathrm{C}\). Describe the effect this would have on the pressure, and, in terms of kinetic theory, explain why this change occurred.

5.161 Power plants driven by fossil-fuel combustion generate substantial greenhouse gases (e.g., \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) ) as well as gases that contribute to poor air quality (e.g., \(\mathrm{SO}_{2}\) ). To evaluate exhaust emissions for regulation purposes, the generally inert nitrogen supplied with air must be included in the balanced reactions; it is further assumed that air is composed of only \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) with a volume/volume ratio of \(3.76: 1.00,\) respectively. In addition to the water produced, the fuel's \(\mathrm{C}\) and \(\mathrm{S}\) are converted to \(\mathrm{CO}_{2}\) and \(\mathrm{SO}_{2}\) at the stack. Given these stipulations, answer the following for a power plant running on a fossil fuel having the formula \(\mathrm{C}_{18} \mathrm{H}_{10} \mathrm{~S}\). Including the \(\mathrm{N}_{2}\) supplied in air, write a balanced combustion equation for the complex fuel assuming \(120 \%\) stoichiometric combustion (i.e., when excess oxygen is present in the product gases). Except for \(\mathrm{N}_{2},\) use only integer coefficients. (See problem \(3.141 .\) ) What gas law is used to effectively convert the given \(\mathrm{N}_{2} / \mathrm{O}_{2}\) volume ratio to a molar ratio when deriving the balanced combustion equation in (a)? (i) Boyle's law (ii) Charles's law (iii) Avogadro's law (iv) ideal gas law c) Assuming the product water condenses, use the result from (b) to determine the stack gas composition on a "dry" basis by calculating the volume/volume percentages for \(\mathrm{CO}_{2}, \mathrm{~N}_{2},\) and \(\mathrm{SO}_{2}\) Assuming the product water remains as vapor, repeat (c) on a "wet" basis.Assuming the product water remains as vapor, repeat (c) on a "wet" basis. Some fuels are cheaper than others, particularly those with higher sulfur contents that lead to poorer air quality. In port, when faced with ever-increasing air quality regulations, large ships operate power plants on more costly, higher-grade fuels; with regulations nonexistent at sea, ships' officers make the cost-saving switch to lower-grade fuels. Suppose two fuels are available, one having the general formula \(\mathrm{C}_{18} \mathrm{H}_{10} \mathrm{~S}\) and the other \(\mathrm{C}_{32} \mathrm{H}_{18} \mathrm{~S} .\) Based on air quality concerns alone, which fuel is more likely to be used when idling in port? Support your answer to (d) by calculating the mass ratio of \(\mathrm{SO}_{2}\) produced to fuel burned for \(\mathrm{C}_{18} \mathrm{H}_{10} \mathrm{~S}\) and \(\mathrm{C}_{32} \mathrm{H}_{18} \mathrm{~S},\) assuming \(120 \%\) stoichiometric combustion.

A volume of air is taken from the earth's surface, at \(19^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\), to the stratosphere, where the temperature is \(-21^{\circ} \mathrm{C}\) and the pressure is \(1.00 \times 10^{-3}\) atm. By what factor is the volume increased?
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