Comprehension # 1
Potash is any potassium mineral that is used for its potassium content. Most of the potash produced in the United States goes into fertilizer. The major sources of potash are potassium choride `(KCI)` and potassium sulphate `(K_(2)SO_(4))`. Potash production is often reported as the potassium oxide `(K_(2)O)` equivalent or the amount of `K_(2)O` that could be made from a given mineral. `KCI` cost `Rs 50 per kg`
What mass `("in kg")` of `K_(2)O` contains the same number of moles of `K` atoms as `1.00 kg KCI` ?

Options

A. `0.158 kg`
B. `0.315 kg`
C. `1.262 kg`
D. `0.631 kg`

Explanation

To solve the problem, we need to find the mass of potassium oxide (K₂O) that contains the same number of moles of potassium (K) atoms as 1.00 kg of potassium chloride (KCl). Let's break this down step by step.

### Step 1: Calculate the molar mass of KCl
- The atomic mass of potassium (K) is approximately 39 g/mol.
- The atomic mass of chlorine (Cl) is approximately 35.5 g/mol.
- Molar mass of KCl = Atomic mass of K + Atomic mass of Cl
\[ \text{Molar mass of KCl} = 39 + 35.5 = 74.5 \text{ g/mol} \]

### Step 2: Calculate the number of moles of KCl in 1 kg
- Convert 1 kg to grams:
\[ 1 \text{ kg} = 1000 \text{ g} \]
- Use the formula for moles:
\[ \text{Moles of KCl} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
\[ \text{Moles of KCl} = \frac{1000 \text{ g}}{74.5 \text{ g/mol}} \approx 13.42 \text{ moles} \]

### Step 3: Determine the number of moles of K atoms in KCl
- Each KCl molecule contains 1 K atom, so the number of moles of K atoms is equal to the number of moles of KCl:
\[ \text{Moles of K} = 13.42 \text{ moles} \]

### Step 4: Calculate the number of moles of K₂O needed
- K₂O contains 2 K atoms per formula unit. Therefore, to find the number of moles of K₂O that would provide the same number of K atoms:
\[ \text{Moles of K₂O} = \frac{\text{Moles of K}}{2} = \frac{13.42}{2} \approx 6.71 \text{ moles} \]

### Step 5: Calculate the molar mass of K₂O
- The atomic mass of oxygen (O) is approximately 16 g/mol.
- Molar mass of K₂O = (2 × atomic mass of K) + atomic mass of O
\[ \text{Molar mass of K₂O} = (2 \times 39) + 16 = 78 + 16 = 94 \text{ g/mol} \]

### Step 6: Calculate the mass of K₂O required
- Use the formula for mass:
\[ \text{Mass of K₂O} = \text{Moles of K₂O} \times \text{Molar mass of K₂O} \]
\[ \text{Mass of K₂O} = 6.71 \text{ moles} \times 94 \text{ g/mol} = 630.67 \text{ g} \]
- Convert grams to kilograms:
\[ \text{Mass of K₂O in kg} = \frac{630.67 \text{ g}}{1000} \approx 0.631 \text{ kg} \]

### Final Answer
The mass of K₂O that contains the same number of moles of K atoms as 1.00 kg of KCl is approximately **0.631 kg**.

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