An oxide of N has vapor density of 23. Find the total number of electrons in its 92 g.(N_(A) = ` Avogadro's number )

Options

A. `46N_(A)`
B. `38N_(A)`
C. `54N_(A)`
D. `30N_(A)`

Explanation

To solve the problem, we need to find the total number of electrons in 92 grams of an oxide of nitrogen with a vapor density of 23. Here’s a step-by-step solution:

### Step 1: Determine the Molecular Mass of the Oxide
The vapor density (VD) is given as 23. The molecular mass (M) can be calculated using the formula:
\[ M = 2 \times \text{Vapor Density} \]
Substituting the given vapor density:
\[ M = 2 \times 23 = 46 \, \text{g/mol} \]

**Hint:** Remember that vapor density is half of the molecular mass for gases.

### Step 2: Identify the Oxide of Nitrogen
The molecular mass of the oxide is 46 g/mol. The general formula for nitrogen oxides can be represented as \( NO_x \). The atomic mass of nitrogen (N) is approximately 14 g/mol, and that of oxygen (O) is approximately 16 g/mol.

Let’s assume the oxide is \( NO_x \):
\[ \text{Molecular mass} = 14 + 16x = 46 \]
Solving for \( x \):
\[ 16x = 46 - 14 = 32 \\ x = \frac{32}{16} = 2 \]
Thus, the oxide is \( NO_2 \).

**Hint:** Use the molecular mass to identify the compound by equating it to the sum of the atomic masses.

### Step 3: Calculate the Number of Electrons in One Molecule of \( NO_2 \)
Now, we need to find the total number of electrons in one molecule of \( NO_2 \):
- Nitrogen (N) has 7 electrons.
- Oxygen (O) has 8 electrons, and since there are 2 oxygen atoms:
\[ \text{Total electrons in } NO_2 = 7 + 2 \times 8 = 7 + 16 = 23 \]

**Hint:** Remember to account for the number of atoms when calculating total electrons.

### Step 4: Calculate the Number of Moles in 92 g of \( NO_2 \)
Using the molecular mass we found earlier (46 g/mol), we can calculate the number of moles in 92 g:
\[ \text{Number of moles} = \frac{\text{Given mass}}{\text{Molecular mass}} = \frac{92 \, \text{g}}{46 \, \text{g/mol}} = 2 \, \text{moles} \]

**Hint:** The number of moles can be calculated by dividing the mass of the substance by its molar mass.

### Step 5: Calculate the Total Number of Molecules
Using Avogadro's number (\( N_A \)), which is approximately \( 6.022 \times 10^{23} \) molecules/mol, we find the total number of molecules:
\[ \text{Total molecules} = \text{Number of moles} \times N_A = 2 \times N_A \]

**Hint:** Avogadro's number is a constant that relates moles to the number of molecules.

### Step 6: Calculate the Total Number of Electrons in 92 g of \( NO_2 \)
Now, we can find the total number of electrons in 92 g of \( NO_2 \):
\[ \text{Total electrons} = \text{Total molecules} \times \text{Electrons per molecule} = (2 \times N_A) \times 23 \]
\[ = 46 N_A \]

**Hint:** Multiply the number of molecules by the number of electrons per molecule to get the total.

### Final Answer
Thus, the total number of electrons in 92 grams of the oxide \( NO_2 \) is \( 46 N_A \).