Problem 53 - General Chemistry: Principles and Modern Applications

Question

A fertilizer contains \(21 \%\) nitrogen by mass. What mass of this fertilizer, in kilograms, is required for an application requiring \(225 \mathrm{g}\) of nitrogen?

Step-by-Step Solution

Answer

The mass of fertilizer required for an application requiring 225 g of nitrogen is approximately 1.07 Kg.

Step 1: Convert Mass of Desired Nitrogen to Kilograms

225 g of nitrogen is equal to \(225 \div 1000 = 0.225\) Kg.

Step 2: Calculate Required Mass of Fertilizer Using Proportions

Given that the fertilizer contains 21% nitrogen by mass, this means that if you have 1kg of the fertilizer, the mass of nitrogen in it would be 0.21Kg (since 21% of 1kg equals 0.21kg). So, to find the mass of the fertilizer that gives 0.225kg of nitrogen, set up a proportion and solve for x (mass of fertilizer). Thus: \( \frac{21}{100} : 1 = 0.225 : x \) , solving for x, yields \( x = \frac{1 \times 0.225}{0.21} \)

Step 3: Final Result

After solving the proportion for x, the desired mass of fertilizer in kilograms is obtained.

Key Concepts

Understanding Proportions in Chemistry

Proportions in chemistry are crucial for solving many problems, especially when dealing with mixtures and solutions. Proportions help us understand the relationship between different components in a chemical mixture. We use them to ensure the right amounts of each element are present, leading to the desired chemical reaction or effect.

In this exercise, to find how much of a certain fertilizer is needed when you know it contains a specific percentage of nitrogen, you set up a proportion. A proportion is essentially a statement that two ratios are equal.

For example, if you know that 1 kilogram of a fertilizer contains 21% nitrogen by mass, you can write this as a ratio: \( \frac{21}{100} \). To find out how much of the fertilizer provides 0.225 kilograms of nitrogen, you set a proportion that connects these two scenarios. Solving proportions lets you find unknown quantities when you know the ratios involved.

Examining Fertilizer Composition

Fertilizer composition refers to the elements or nutrients that make up a fertilizer. These compositions are often presented in percentages by mass for key nutrients like nitrogen (N), phosphorus (P), and potassium (K), usually in the form of an "N-P-K" ratio.

This labeling standard makes it easy to understand what percentage of each nutrient is in the fertilizer. For instance:

• An "N-P-K" ratio of 21-0-0 means 21% nitrogen, 0% phosphorus, and 0% potassium.

In our exercise, we analyzed a fertilizer with 21% nitrogen. Understanding this percentage helps determine how much fertilizer is necessary to provide a specific amount of nitrogen.

When choosing a fertilizer, knowing its composition is vital for fulfilling the specific nutrient requirements of plants, ensuring healthy growth.

Calculating Nitrogen Percentage

Calculating the nitrogen percentage within a fertilizer involves using given data to understand how much nitrogen is present in a total mass.

In our problem, we knew that 21% of the fertilizer’s mass is nitrogen. To find out how much fertilizer you need for a certain amount of nitrogen, knowing how to manipulate percentages and convert between units is important.

Steps to calculate this include:

• Understanding that 21% nitrogen means every 100 kilograms of fertilizer contains 21 kilograms of nitrogen.
• Converting any given nitrogen requirements from grams to kilograms as needed (since percentages are often based on kilograms).
• Setting up the proportion to solve for the total fertilizer mass required to meet the nitrogen needs (using the equation \( x = \frac{1 \times 0.225}{0.21} \)).

This calculation ensures the correct amount of nutrient is added without excess, enabling optimal plant growth.