Problem 87 - Chemistry: Principles and Reactions

Question

All the fertilizers listed below contribute nitrogen to the soil. If all these fertilizers are sold for the same price per gram of nitrogen, which will cost the least per 50 -lb bag? urea, \(\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}\) ammonia, \(\mathrm{NH}_{3}\) ammonium nitrate, \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) guanidine, \(\mathrm{HNC}\left(\mathrm{NH}_{2}\right)_{2}\)

Step-by-Step Solution

Answer

Answer: The ammonium nitrate fertilizer will cost the least per 50-lb bag, with a cost of $17.50.

Step 1: Calculate molecular weight of each fertilizer

Use the periodic table to find the atomic weights of the elements within each fertilizer, and then add these weights together to obtain the molecular weight of the compound. Molecular weight of urea: $$(\mathrm{NH}_{2})_{2}\mathrm{CO} = 2(\mathrm{N} + 2\mathrm{H}) + \mathrm{C} + \mathrm{O} = 2(14 + 2) + 12 + 16 = 60$$ Molecular weight of ammonia: $$\mathrm{NH}_{3} = \mathrm{N} + 3\mathrm{H} = 14 + 3 = 17$$ Molecular weight of ammonium nitrate: $$\mathrm{NH}_{4}\mathrm{NO}_{3} = \mathrm{N} + 4\mathrm{H} + \mathrm{N} + \mathrm{O} + 2\mathrm{O} = 2\mathrm{N} + 4\mathrm{H} + 3\mathrm{O} = 2(14) + 4 + 3(16) = 80$$ Molecular weight of guanidine: $$\mathrm{HNC}\left(\mathrm{NH}_{2}\right)_{2} = \mathrm{H} + \mathrm{N} + \mathrm{C} + 2(\mathrm{N} + 2\mathrm{H}) = 1 + 14 + 12 + 2(14 + 2) = 59$$

Step 2: Find the number of nitrogen atoms in each fertilizer molecule

Count the nitrogen atoms in each fertilizer molecule: Urea: 2 nitrogen atoms Ammonia: 1 nitrogen atom Ammonium nitrate: 2 nitrogen atoms Guanidine: 3 nitrogen atoms

Step 3: Calculate the weight of nitrogen in each fertilizer

Multiply the number of nitrogen atoms by the molecular weight of nitrogen (14) and divide by the molecular weight of the fertilizer compound to find the weight of nitrogen content in each fertilizer: Urea: \(2(14) / 60 = 0.467\) Ammonia: \(1(14) / 17 = 0.824\) Ammonium nitrate: \(2(14) / 80 = 0.350\) Guanidine: \(3(14) / 59 = 0.712\)

Step 4: Calculate the cost of each fertilizer per 50-lb bag

Multiply the weight of nitrogen content in each fertilizer (in lb) by the price per gram of nitrogen: Cost of urea per 50-lb bag: \(50(0.467) = 23.35\) Cost of ammonia per 50-lb bag: \(50(0.824) = 41.20\) Cost of ammonium nitrate per 50-lb bag: \(50(0.350) = 17.50\) Cost of guanidine per 50-lb bag: \(50(0.712) = 35.60\) The ammonium nitrate fertilizer will cost the least per 50-lb bag, as its cost is \(17.50\).

Key Concepts

Fertilizers

Fertilizers are essential materials used to boost the fertility of the soil, ensuring that plants receive the adequate nutrients necessary for healthy growth. Each type of fertilizer is composed of different molecules that contribute essential elements such as nitrogen, phosphorus, and potassium to the soil. Nitrogen, in particular, is crucial for the growth of plants because it is a primary component of amino acids, the building blocks of proteins. These proteins play a significant role in the general health and growth of plants by supporting the process of photosynthesis and the development of chlorophyll.

When discussing fertilizers, it's essential to consider types that mainly supply nitrogen because nitrogen directly affects plant productivity. Various compounds such as urea, ammonia, ammonium nitrate, and guanidine are primarily utilized because of their high nitrogen content, represented as \((\mathrm{NH}_2)_2\mathrm{CO}\), \(\mathrm{NH}_3\), \(\mathrm{NH}_4\mathrm{NO}_3\), and \(\mathrm{HNC}(\mathrm{NH}_2)_2\), respectively. Understanding the molecular structure of these compounds helps us determine how efficiently they can provide nitrogen for plant absorption.

Nitrogen Content

The nitrogen content of a fertilizer is a crucial factor to consider as it determines how much nitrogen can be supplied to the plants. Every fertilizer type has a specific number of nitrogen atoms within its molecular structure. For example, urea and ammonium nitrate each have two nitrogen atoms, whereas ammonia has only one, and guanidine includes three nitrogen atoms.

To determine a fertilizer's true nitrogen content, we calculate the weight of nitrogen atoms in relation to the molecular weight of the fertilizer compound. This is done by multiplying the number of nitrogen atoms by the atomic weight of nitrogen, which is 14, and dividing by the compound's total molecular weight.

• Urea has a nitrogen content of approximately 46.7%.
• Ammonia provides around 82.4% nitrogen content.
• Ammonium nitrate offers approximately 35%.
• Lastly, guanidine presents roughly 71.2% nitrogen content.

The higher the nitrogen content, the more efficient the fertilizer is at providing nitrogen to plants, influencing the decision on which fertilizer to use depending on the specific crop's needs.

Cost Analysis

Conducting a cost analysis of fertilizers can directly affect budgeting decisions for agricultural projects. Given the same price per gram of nitrogen, it’s more economical to choose the fertilizer that offers the highest nitrogen content per kilogram. This analysis involves calculating the cost of each fertilizer per 50-pound bag, taking into account how much nitrogen each type supplies.

We find the cost by multiplying the nitrogen weight percentage by the total weight of the fertilizer bag. Based on the calculations, here's a breakdown:

• The cost for a 50-lb bag of urea is approximately $23.35.
• Ammonia, being rich in nitrogen, costs about $41.20, making it pricier.
• Ammonium nitrate, despite a lower nitrogen percentage, is the cheapest at $17.50 per bag.
• Guanidine comes to $35.60 for a 50-lb bag.

Hence, ammonium nitrate offers a cost advantage when nitrogen is priced the same per gram, making it the most cost-effective option for farmers looking to optimize their nitrogen purchase for plant fertilization.