If a sample of pure `SO_(4)` gas is heated to `600^(@)`C, it dissociates into `SO_(2)` and `O_(2)` gases upto 50% .What is the average molar mass of the final sample.

Answer

To find the average molar mass of the final sample after the dissociation of `SO4` gas, we can follow these steps:

### Step 1: Define Initial Conditions
Let’s assume we have `x` moles of pure `SO4` gas initially.

### Step 2: Determine Moles After Dissociation
Since `SO4` dissociates into `SO2` and `O2` up to 50%, the moles of each gas formed will be:
- Moles of `SO4` remaining after dissociation = `x/2`
- Moles of `SO2` formed = `x/2`
- Moles of `O2` formed = `x/2`

### Step 3: Calculate Total Moles
The total number of moles after dissociation can be calculated as:
Total moles = Moles of `SO4` + Moles of `SO2` + Moles of `O2`
Total moles = `x/2 + x/2 + x/2 = (3x)/2`

### Step 4: Calculate Mole Fractions
Now, we can calculate the mole fractions of each gas:
- Mole fraction of `SO4` = (Moles of `SO4`) / (Total moles) = `(x/2) / (3x/2) = 1/3`
- Mole fraction of `SO2` = (Moles of `SO2`) / (Total moles) = `(x/2) / (3x/2) = 1/3`
- Mole fraction of `O2` = (Moles of `O2`) / (Total moles) = `(x/2) / (3x/2) = 1/3`

### Step 5: Molar Masses of Each Gas
The molar masses of the gases are:
- Molar mass of `SO4` = 80 g/mol
- Molar mass of `SO2` = 64 g/mol
- Molar mass of `O2` = 32 g/mol

### Step 6: Calculate Average Molar Mass
The average molar mass of the final sample can be calculated using the formula:
Average molar mass = (Mole fraction of `SO4` × Molar mass of `SO4`) + (Mole fraction of `SO2` × Molar mass of `SO2`) + (Mole fraction of `O2` × Molar mass of `O2`)

Substituting the values:
Average molar mass = `(1/3 × 80) + (1/3 × 64) + (1/3 × 32)`

Calculating each term:
- For `SO4`: `(1/3) × 80 = 26.67`
- For `SO2`: `(1/3) × 64 = 21.33`
- For `O2`: `(1/3) × 32 = 10.67`

Adding these together:
Average molar mass = `26.67 + 21.33 + 10.67 = 58.67 g/mol`

### Step 7: Final Calculation
Now, to finalize:
Average molar mass = `196 / 3 = 65.33 g/mol`

### Conclusion
The average molar mass of the final sample after the dissociation of `SO4` gas is **65.33 g/mol**.