A mixture of `NO_(2)` and `N_(2)O_(4)` has a vapor density of `38.3` at 300 K. What is the number of moles of `NO_(2)` in 100 g of themixture ?

Options

A. 0.043
B. 4.4
C. 3.4
D. 3.86

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Molecular Weight of the Mixture
The vapor density (VD) of the mixture is given as 38.3. The molecular weight (MW) can be calculated using the formula:
\[ \text{Molecular Weight} = \text{Vapor Density} \times 2 \]
Substituting the value:
\[ \text{Molecular Weight} = 38.3 \times 2 = 76.6 \, \text{g/mol} \]

### Step 2: Calculate the Total Moles of the Mixture
We have 100 grams of the mixture. The number of moles (n) of the mixture can be calculated using the formula:
\[ n = \frac{\text{mass}}{\text{molecular weight}} \]
Substituting the values:
\[ n = \frac{100 \, \text{g}}{76.6 \, \text{g/mol}} \approx 1.30 \, \text{moles} \]

### Step 3: Set Up the Mass Relationships
Let the mass of \(NO_2\) be \(x\) grams. Then the mass of \(N_2O_4\) will be \(100 - x\) grams. The molecular weights are:
- \(NO_2\): 46 g/mol
- \(N_2O_4\): 92 g/mol (since \(N_2O_4\) consists of two \(NO_2\) units)

### Step 4: Write the Equation for Total Moles
The total moles of the mixture can be expressed as:
\[ \frac{x}{46} + \frac{100 - x}{92} = 1.30 \]

### Step 5: Solve the Equation
To solve the equation, we will first find a common denominator:
\[ \frac{x}{46} + \frac{100 - x}{92} = 1.30 \]
The common denominator is \(92\):
\[ \frac{2x}{92} + \frac{100 - x}{92} = 1.30 \]
Combining the terms:
\[ \frac{2x + 100 - x}{92} = 1.30 \]
This simplifies to:
\[ \frac{x + 100}{92} = 1.30 \]
Multiplying both sides by 92:
\[ x + 100 = 119.6 \]
Now, solving for \(x\):
\[ x = 119.6 - 100 = 19.6 \, \text{grams} \]

### Step 6: Calculate the Moles of \(NO_2\)
Now, we can find the number of moles of \(NO_2\):
\[ \text{Moles of } NO_2 = \frac{x}{46} = \frac{19.6}{46} \approx 0.426 \, \text{moles} \]

### Final Answer
The number of moles of \(NO_2\) in 100 g of the mixture is approximately **0.426 moles**.