A saturated hydrocarbon having molecular formula `C_(n)H_(2n+2)` diffuses through a porous membrane twice as fast as sulpjur dioxide.
Determine the molecualr mass of hydrocarbon and name it.

Explanation

To determine the molecular mass of the hydrocarbon and name it, we can follow these steps:

### Step 1: Understand the relationship between diffusion rates and molecular mass
According to Graham's law of effusion, the rate of diffusion of a gas is inversely proportional to the square root of its molecular mass. This can be expressed as:

\[ \frac{R_1}{R_2} = \sqrt{\frac{M_2}{M_1}} \]

where \( R_1 \) and \( R_2 \) are the rates of diffusion of the two gases, and \( M_1 \) and \( M_2 \) are their respective molecular masses.

### Step 2: Set up the equation based on the problem
In this case, we know that the hydrocarbon diffuses twice as fast as sulfur dioxide (SO2). Therefore, we can set:

\[ \frac{R_{alkane}}{R_{SO2}} = 2 \]

Let \( M_{SO2} \) be the molecular mass of sulfur dioxide, which is calculated as follows:

\[ M_{SO2} = 32 \, \text{(S)} + 16 \, \text{(O)} \times 2 = 32 + 32 = 64 \, \text{g/mol} \]

### Step 3: Substitute into Graham's law
Now substituting the known values into Graham's law:

\[ 2 = \sqrt{\frac{64}{M_{alkane}}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:

\[ 4 = \frac{64}{M_{alkane}} \]

### Step 5: Rearrange to find the molecular mass of the alkane
Rearranging the equation to solve for \( M_{alkane} \):

\[ M_{alkane} = \frac{64}{4} = 16 \, \text{g/mol} \]

### Step 6: Determine the molecular formula of the hydrocarbon
The molecular formula for a saturated hydrocarbon (alkane) is given by:

\[ C_nH_{2n+2} \]

### Step 7: Set up the equation for molecular mass
The molecular mass can also be expressed as:

\[ M_{alkane} = 12n + (2n + 2) \cdot 1 = 12n + 2n + 2 = 14n + 2 \]

Setting this equal to the molecular mass we found:

\[ 14n + 2 = 16 \]

### Step 8: Solve for \( n \)
Subtract 2 from both sides:

\[ 14n = 14 \]

Dividing both sides by 14 gives:

\[ n = 1 \]

### Step 9: Write the molecular formula
Substituting \( n \) back into the formula gives:

\[ C_1H_{2(1)+2} = CH_4 \]

### Step 10: Name the hydrocarbon
The hydrocarbon with the formula \( CH_4 \) is known as methane.

### Final Answer
The molecular mass of the hydrocarbon is 16 g/mol, and the hydrocarbon is methane.

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