A mixture of methane and ethane in the molar ratio of x:y has a mean molar mass of 20. what would be the mean molar mass, if the gases are mixed in the molar ratio of y:x?

Options

A. 20 u
B. 25 u
C. 24 u
D. 15 u

Answer

To solve the problem step by step, we will analyze the given information and derive the mean molar mass when the gases are mixed in the opposite ratio.

### Step 1: Understand the Molar Masses
The molar masses of the gases involved are:
- Molar mass of methane (CH₄) = 16 g/mol
- Molar mass of ethane (C₂H₆) = 30 g/mol

### Step 2: Set Up the Initial Equation
Given that the molar ratio of methane to ethane is x:y and the mean molar mass of the mixture is 20 g/mol, we can express this as:
\[ \frac{16x + 30y}{x + y} = 20 \]

### Step 3: Rearranging the Equation
Multiply both sides by (x + y):
\[ 16x + 30y = 20(x + y) \]
This simplifies to:
\[ 16x + 30y = 20x + 20y \]

### Step 4: Collect Like Terms
Rearranging the equation gives:
\[ 30y - 20y = 20x - 16x \]
This simplifies to:
\[ 10y = 4x \]
From this, we can find the relationship between x and y:
\[ y = \frac{2}{5}x \]

### Step 5: Set Up the New Molar Ratio
Now, we need to find the mean molar mass when the gases are mixed in the molar ratio of y:x. Using the relationship we found, we can substitute y:
\[ y = \frac{2}{5}x \]
Thus, the new ratio y:x is \(\frac{2}{5}x : x\).

### Step 6: Express the New Mean Molar Mass
The mean molar mass for the new ratio can be expressed as:
\[ \text{Mean Molar Mass} = \frac{16y + 30x}{y + x} \]
Substituting y:
\[ = \frac{16\left(\frac{2}{5}x\right) + 30x}{\frac{2}{5}x + x} \]

### Step 7: Simplify the Expression
Calculating the numerator:
\[ = \frac{\frac{32}{5}x + 30x}{\frac{2}{5}x + \frac{5}{5}x} = \frac{\frac{32}{5}x + \frac{150}{5}x}{\frac{7}{5}x} \]
This simplifies to:
\[ = \frac{\frac{182}{5}x}{\frac{7}{5}x} \]
The x terms cancel out:
\[ = \frac{182}{7} \]

### Step 8: Calculate the Final Mean Molar Mass
Calculating \(\frac{182}{7}\):
\[ = 26 \text{ g/mol} \]

### Final Answer
The mean molar mass when the gases are mixed in the molar ratio of y:x is **26 g/mol**.

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