Question

The nitrogen in sodium nitrate and in ammonium sulfate is available to plants as fertilizer. Which is the more economical source of nitrogen, a fertilizer containing 30.0\(\%\) sodium nitrate by weight and costing \(\$ 9.00\) per 100 lb or one containing 20.0\(\%\) ammonium sulfate by weight and costing \(\$ 8.10\) per 100 \(\mathrm{lb} ?\)

Short answer

Sodium nitrate is more economical at \$ 0.30 per lb of nitrogen, compared to ammonium sulfate at \$ 0.405 per lb of nitrogen.

Step-by-step solution

Calculate the cost of nitrogen per pound for sodium nitrate

First, determine the amount of nitrogen in 100 lb of sodium nitrate. Since the nitrogen content is 30.0%, there are 30.0 lb of nitrogen in 100 lb of sodium nitrate. The cost per pound for the nitrogen in sodium nitrate is calculated by dividing the total cost by the amount of nitrogen. So, it is \(\frac{\$ 9.00}{30.0\,\mathrm{lb}} = \$ 0.30\,\mathrm{per\,lb}\,\mathrm{of\,nitrogen}\).

Calculate the cost of nitrogen per pound for ammonium sulfate

Next, calculate the amount of nitrogen in 100 lb of ammonium sulfate. The nitrogen content is 20.0%, so there are 20.0 lb of nitrogen in 100 lb of ammonium sulfate. The cost per pound for the nitrogen in ammonium sulfate is found by dividing the total cost by the amount of nitrogen. Hence, \(\frac{\$ 8.10}{20.0\,\mathrm{lb}} = \$ 0.405\,\mathrm{per\,lb}\,\mathrm{of\,nitrogen}\).

Compare the costs to determine the more economical source

The final step is to compare the cost of nitrogen per pound for both fertilizers. Sodium nitrate costs \(\$ 0.30\,\mathrm{per\,lb}\) of nitrogen, while ammonium sulfate costs \(\$ 0.405\,\mathrm{per\,lb}\) of nitrogen. Since the cost per pound of nitrogen is lower for sodium nitrate, it is the more economical source of nitrogen for plants as fertilizer.

Key concepts

Sodium Nitrate for Plants

Sodium nitrate is a type of nitrogenous fertilizer that can be directly utilized by plants to promote growth and improve yield. Often abbreviated as NaNO3, it is an inorganic compound combining sodium, nitrogen, and oxygen. For plants, nitrogen is a crucial element as it's a component of chlorophyll, the substance that grants plants their green color and is integral to photosynthesis.

Because sodium nitrate has a relatively high nitrogen content, it is particularly beneficial to plant growth. This type of fertilizer is easily soluble in water, meaning it can quickly deliver nutrients to plants. Additionally, sodium nitrate is less acidic than some other nitrogen fertilizers, which can be advantageous for managing soil pH levels. However, due to its high solubility, it can also leach out from the soil if over-applied or applied before heavy rainfall.

When using sodium nitrate as a fertilizer, it's essential to consider its concentration and application rate. Overuse can lead to a toxic buildup of nitrates in the soil, harming both plant growth and soil health.

Ammonium Sulfate as Fertilizer

Ammonium sulfate, denoted as (NH4)2SO4, serves as another popular nitrogen fertilizer, providing both nitrogen and sulfur nutrients to plants. The ammonium ion (NH4+) serves as a source of nitrogen, whereas the sulfate ion (SO4--) supplies sulfur, which is also necessary for plant growth.

Ammonium sulfate is known for its ability to maintain the acidity of the soil, or lower the pH, which can be extremely beneficial for acid-loving plants. Its nitrogen is slowly released, which makes it less likely to leach out from the soil. This slow-release property helps to provide a sustained supply of nitrogen over the growing season.

However, its acidifying effect means it might not be suitable for all soil types or plants that prefer a neutral to alkaline soil pH. It is invariably important for gardeners and farmers to test their soil before applying ammonium sulfate to ensure it benefits their specific soil conditions and plant requirements.

Nitrogen Content Calculation

Calculating the nitrogen content of fertilizers is crucial in ensuring that plants get adequate nutrition without the risk of over-fertilization that can damage the plants and the environment. Farmers and gardeners use this calculation to determine the most economical choice for their specific needs.

To calculate the nitrogen content in fertilizers, as exemplified in the exercise, one must know the percentage of nitrogen present. This percentage is multiplied by the weight of the fertilizer to find the amount of nitrogen available. The cost efficiency is then determined by dividing the total cost of the fertilizer by the amount of nitrogen it contains, yielding the cost per pound of nitrogen.

For instance, in the provided solution, sodium nitrate has a 30.0% nitrogen content, thus 30.0 lb of nitrogen per 100 lb of the fertilizer. Given the cost, you calculate the price per pound of nitrogen, which facilitates a direct comparison with other nitrogen sources. This approach to calculating nitrogen content and cost efficiency ensures that the user can compare different fertilizers and make a decision that balances cost with nutritional benefit for the plants.