A sample of `C_(x)O_(y)` having `3xx10^(19)` molecules is weighing `3.4 mg`. Then the value of x and y is respectively …………… `(Take N_(A) = 6xx10^(23))`

Options

A. `1,2`
B. `3,2`
C. `2,2`
D. `5,2`

Explanation

To solve the problem, we need to find the values of x and y in the molecular formula \( C_xO_y \) given the number of molecules and the mass of the sample. Here’s a step-by-step solution:

### Step 1: Calculate the number of moles from the number of molecules.
We are given the number of molecules as \( 3 \times 10^{19} \) and Avogadro's number \( N_A = 6 \times 10^{23} \).

Using the formula for moles:
\[ \text{Moles} = \frac{\text{Number of molecules}}{N_A} \]
Substituting the values:
\[ \text{Moles} = \frac{3 \times 10^{19}}{6 \times 10^{23}} = 0.5 \times 10^{-4} = 5 \times 10^{-5} \text{ moles} \]

### Step 2: Convert the mass from milligrams to grams.
The mass of the sample is given as \( 3.4 \text{ mg} \). To convert this to grams:
\[ 3.4 \text{ mg} = 3.4 \times 10^{-3} \text{ g} \]

### Step 3: Use the relationship between moles, mass, and molar mass.
The formula relating moles, mass, and molar mass is:
\[ \text{Moles} = \frac{\text{Mass (g)}}{\text{Molar Mass (g/mol)}} \]
Let the molar mass be \( M \). Substituting the known values:
\[ 5 \times 10^{-5} = \frac{3.4 \times 10^{-3}}{M} \]
Rearranging this gives:
\[ M = \frac{3.4 \times 10^{-3}}{5 \times 10^{-5}} = 68 \text{ g/mol} \]

### Step 4: Set up the equation for the molar mass of \( C_xO_y \).
The molar mass of the compound can be expressed as:
\[ M = 12x + 16y \]
Setting this equal to the calculated molar mass:
\[ 12x + 16y = 68 \]

### Step 5: Solve for x and y using trial and error.
We can try different integer values for \( x \) and \( y \) that satisfy the equation \( 12x + 16y = 68 \).

1. **Let \( x = 2 \)**:
\[ 12(2) + 16y = 68 \implies 24 + 16y = 68 \implies 16y = 44 \implies y = 2.75 \quad \text{(not an integer)} \]

2. **Let \( x = 3 \)**:
\[ 12(3) + 16y = 68 \implies 36 + 16y = 68 \implies 16y = 32 \implies y = 2 \quad \text{(integer)} \]

Thus, we find \( x = 3 \) and \( y = 2 \).

### Final Answer:
The values of \( x \) and \( y \) are \( 3 \) and \( 2 \) respectively.

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