Nitrogen (N), phosporus (P) and potassium (K) are the main nutrients in plant fertilizers. . According to an industry convention , the numbers on the label refer to the mas % of N, `P_(2)O_(5)` and `K_(2)O` in that order. If the N:P:K (in terms of moles of each elements) ratio of a fertilizer labelled as 28:14.2:4.7 is expressed as 20:x:1.0, then calculate x.

Answer

To solve the problem of finding the value of \( x \) in the N:P:K ratio expressed as \( 20:x:1.0 \) from the fertilizer label of \( 28:14.2:4.7 \), we will follow these steps:

### Step 1: Understand the Mass Percentages
The fertilizer label indicates:
- Nitrogen (N) = 28%
- Phosphorus (as \( P_2O_5 \)) = 14.2%
- Potassium (as \( K_2O \)) = 4.7%

### Step 2: Convert Mass Percentages to Moles
We need to convert the mass percentages to moles using their molar masses:
- Molar mass of Nitrogen (N) = 14 g/mol
- Molar mass of \( P_2O_5 \) = \( 2 \times 31 + 5 \times 16 = 62 + 80 = 142 \) g/mol
- Molar mass of \( K_2O \) = \( 2 \times 39 + 16 = 78 + 16 = 94 \) g/mol

Now, we can calculate the moles of each nutrient based on the mass percentages.

### Step 3: Calculate Moles of Each Nutrient
1. **Moles of Nitrogen (N)**:
\[ \text{Moles of N} = \frac{28 \text{ g}}{14 \text{ g/mol}} = 2 \text{ moles} \]

2. **Moles of Phosphorus (P)**:
Since \( P_2O_5 \) contains 2 moles of phosphorus:
\[ \text{Moles of } P_2O_5 = \frac{14.2 \text{ g}}{142 \text{ g/mol}} = 0.1 \text{ moles} \]
Therefore, moles of Phosphorus (P):
\[ \text{Moles of P} = 2 \times 0.1 = 0.2 \text{ moles} \]

3. **Moles of Potassium (K)**:
Since \( K_2O \) contains 2 moles of potassium:
\[ \text{Moles of } K_2O = \frac{4.7 \text{ g}}{94 \text{ g/mol}} \approx 0.05 \text{ moles} \]
Therefore, moles of Potassium (K):
\[ \text{Moles of K} = 2 \times 0.05 = 0.1 \text{ moles} \]

### Step 4: Establish the Ratio
Now we have the moles:
- Moles of N = 2
- Moles of P = 0.2
- Moles of K = 0.1

To express this in the form \( 20:x:1.0 \), we need to find the ratio of N:P:K.

### Step 5: Normalize the Ratios
To express the ratio in terms of \( 20:x:1.0 \):
- For N = 2, we can scale it to 20:
\[ \text{Scale factor} = \frac{20}{2} = 10 \]
- For P:
\[ x = 0.2 \times 10 = 2 \]
- For K:
\[ 1.0 = 0.1 \times 10 \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{2} \]