Nitrogen (N), Phosphorus (P), and Potassium (K) are the main nutrients in plant fertilizers. According to an industry convention, the numbers on the label refer to the mass percents of N, P2O5, and K2O, in that order. Calculate the N:P:K ratio of a 30:10:10 fertilizer in terms of moles of each element, and express it as X:y:1.0.

To determine the N:P:K ratio of a 30:10:10 fertilizer in terms of moles of nitrogen (N), phosphorus (P), and potassium (K), we first need to convert the percentages of each nutrient into moles. The numbers on the fertilizer label represent the mass percentages of nitrogen, phosphorus pentoxide (P2O5), and potassium oxide (K2O). Let's break this down step by step.

Step 1: Understanding the Fertilizer Composition

The fertilizer label indicates:

• Nitrogen (N): 30%
• Phosphorus (P2O5): 10%
• Potassium (K2O): 10%

Step 2: Converting Mass Percentages to Moles

To find the mole ratio, we need to convert the mass percentages into moles using the molar masses of each compound:

• Molar mass of N = 14.01 g/mol
• Molar mass of P2O5 = 2(30.97) + 5(16.00) = 61.94 + 80.00 = 141.94 g/mol
• Molar mass of K2O = 2(39.10) + 16.00 = 78.20 + 16.00 = 94.20 g/mol

Step 3: Calculating Moles from Percentages

Assuming we have 100 grams of the fertilizer, we can calculate the mass of each nutrient:

• Mass of N = 30 g
• Mass of P2O5 = 10 g
• Mass of K2O = 10 g

Now, we convert these masses into moles:

• Moles of N = 30 g / 14.01 g/mol ≈ 2.14 moles
• Moles of P2O5 = 10 g / 141.94 g/mol ≈ 0.0704 moles
• Moles of K2O = 10 g / 94.20 g/mol ≈ 0.106 moles

Step 4: Converting P2O5 to P

Since we need the ratio in terms of elemental phosphorus (P), we must convert the moles of P2O5 to moles of P. The conversion is based on the formula:

1 mole of P2O5 contains 2 moles of P.

Thus, moles of P = 0.0704 moles P2O5 × 2 = 0.1408 moles of P.

Step 5: Finding the Ratio

Now we have:

• N: 2.14 moles
• P: 0.1408 moles
• K: 0.106 moles

To express this as a ratio of X:y:1.0, we first need to find the smallest number of moles among N, P, and K, which is 0.106 moles (for K). We divide each value by 0.106:

• N: 2.14 / 0.106 ≈ 20.19
• P: 0.1408 / 0.106 ≈ 1.32
• K: 0.106 / 0.106 = 1.0

Step 6: Finalizing the Ratio

To express this in a simple ratio format, we can round the values:

• N: 20
• P: 1.32 (approximately 1.3)
• K: 1

Thus, the N:P:K ratio can be expressed as:

20:1.3:1

In conclusion, the N:P:K ratio of a 30:10:10 fertilizer, expressed in terms of moles, is approximately 20:1.3:1. This ratio indicates the relative amounts of each nutrient available to plants, which is crucial for understanding how to effectively use fertilizers in agriculture.