Under same conditions of temperature and pressure, a hydrocarbon of molecular formula `C_(n)H_(2n-2)` was found to diffuse `3sqrt3` times slower than hydrogen. Find the value of n.

Options

A. n = 2
B. n = 4
C. n = 3
D. n = 1

Explanation

To solve the problem step by step, we will use Graham's law of diffusion and the information provided about the hydrocarbon.

### Step 1: Understand the relationship given in the problem
We know that the hydrocarbon with the formula \( C_nH_{2n-2} \) diffuses \( 3\sqrt{3} \) times slower than hydrogen. According to Graham's law of diffusion, the rate of diffusion is inversely proportional to the square root of the molecular mass.

### Step 2: Set up the equation using Graham's law
Let \( R_{H_2} \) be the rate of diffusion of hydrogen and \( R_{hydrocarbon} \) be the rate of diffusion of the hydrocarbon. From the problem, we have:
\[ \frac{R_{H_2}}{R_{hydrocarbon}} = 3\sqrt{3} \]

### Step 3: Relate the rates to molecular masses
According to Graham's law:
\[ \frac{R_{H_2}}{R_{hydrocarbon}} = \sqrt{\frac{M_{hydrocarbon}}{M_{H_2}}} \]
Where \( M_{H_2} \) is the molecular mass of hydrogen, which is 2 g/mol. Therefore, we can set up the equation:
\[ 3\sqrt{3} = \sqrt{\frac{M_{hydrocarbon}}{2}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (3\sqrt{3})^2 = \frac{M_{hydrocarbon}}{2} \]
\[ 27 = \frac{M_{hydrocarbon}}{2} \]

### Step 5: Solve for the molecular mass of the hydrocarbon
Multiplying both sides by 2:
\[ M_{hydrocarbon} = 54 \text{ g/mol} \]

### Step 6: Write the expression for the molecular mass of the hydrocarbon
The molecular mass of the hydrocarbon \( C_nH_{2n-2} \) can be calculated as follows:
\[ M_{hydrocarbon} = 12n + (2n - 2) \]
This simplifies to:
\[ M_{hydrocarbon} = 12n + 2n - 2 = 14n - 2 \]

### Step 7: Set up the equation with the known molecular mass
Now we set the expression for the molecular mass equal to the value we found:
\[ 14n - 2 = 54 \]

### Step 8: Solve for \( n \)
Adding 2 to both sides:
\[ 14n = 56 \]
Dividing both sides by 14:
\[ n = 4 \]

### Final Answer
The value of \( n \) is \( 4 \).

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