Nitrogen (N), phosporus(P), and potassium (K) are the main nutrients in the plant fertilizers. According to an industry convention , the numbers on the label refer to the mass % of `N_2` ,`P_(2)O`, and `K_(2)O`, in that order. Calculate the N : P : K ratio of a 28 :14.2 : 4.7 fertilizer in terms of moles of each elements, and express it as x : y :1.0. (P = 31 , K = 39 )

Answer

To solve the problem of calculating the N : P : K ratio of a fertilizer labeled as 28 : 14.2 : 4.7 in terms of moles of each element, we can follow these steps:

### Step 1: Calculate the molar mass of each compound
1. **For \(N_2\)**:
- Molar mass of nitrogen (N) = 14 g/mol
- Molar mass of \(N_2 = 14 \times 2 = 28 \, \text{g/mol}\)

2. **For \(P_2O_5\)**:
- Molar mass of phosphorus (P) = 31 g/mol
- Molar mass of oxygen (O) = 16 g/mol
- Molar mass of \(P_2O_5 = (31 \times 2) + (16 \times 5) = 62 + 80 = 142 \, \text{g/mol}\)

3. **For \(K_2O\)**:
- Molar mass of potassium (K) = 39 g/mol
- Molar mass of \(K_2O = (39 \times 2) + 16 = 78 + 16 = 94 \, \text{g/mol}\)

### Step 2: Convert the mass percentages to moles
Using the mass percentages given (28% for \(N_2\), 14.2% for \(P_2O_5\), and 4.7% for \(K_2O\)), we can calculate the moles of each component:

1. **Moles of \(N_2\)**:
\[ \text{Moles of } N_2 = \frac{28 \, \text{g}}{28 \, \text{g/mol}} = 1.0 \, \text{mol} \]

2. **Moles of \(P_2O_5\)**:
\[ \text{Moles of } P_2O_5 = \frac{14.2 \, \text{g}}{142 \, \text{g/mol}} \approx 0.1 \, \text{mol} \]

3. **Moles of \(K_2O\)**:
\[ \text{Moles of } K_2O = \frac{4.7 \, \text{g}}{94 \, \text{g/mol}} \approx 0.05 \, \text{mol} \]

### Step 3: Convert moles of compounds to moles of elements
1. **Moles of Nitrogen (N)**:
- Since \(N_2\) contains 2 moles of N:
\[ \text{Moles of } N = 1.0 \, \text{mol} \times 2 = 2.0 \, \text{mol} \]

2. **Moles of Phosphorus (P)**:
- Since \(P_2O_5\) contains 2 moles of P:
\[ \text{Moles of } P = 0.1 \, \text{mol} \times 2 = 0.2 \, \text{mol} \]

3. **Moles of Potassium (K)**:
- Since \(K_2O\) contains 2 moles of K:
\[ \text{Moles of } K = 0.05 \, \text{mol} \times 2 = 0.1 \, \text{mol} \]

### Step 4: Establish the ratio of N : P : K
Now we have:
- Moles of N = 2.0
- Moles of P = 0.2
- Moles of K = 0.1

To express this as a ratio \(x : y : 1.0\), we can divide each mole value by the moles of K (0.1):

1. **N ratio**:
\[ \frac{2.0}{0.1} = 20 \]

2. **P ratio**:
\[ \frac{0.2}{0.1} = 2 \]

3. **K ratio**:
\[ \frac{0.1}{0.1} = 1 \]

Thus, the ratio of N : P : K can be expressed as:
\[ 20 : 2 : 1 \]

### Final Answer
The N : P : K ratio in terms of moles is:
\[ 20 : 2 : 1 \]