Calculate percentage change in `M_(avg)` of the mixture, if `PCl_(5)` undergo `50%` decomposition.
`PCl_(5) rArr PCl_(3) + Cl_(2)`

Options

A. 0.5
B. 0.6666
C. 0.3333
D. zero

Answer

To calculate the percentage change in the average molar mass (M_avg) of the mixture when PCl₅ undergoes 50% decomposition, we can follow these steps:

### Step 1: Define Initial Conditions
Assume we start with 2n moles of PCl₅. The molar mass of PCl₅ is approximately 208.24 g/mol.

### Step 2: Calculate Initial Molar Mass (M_avg_initial)
The initial average molar mass (M_avg_initial) of the mixture is simply the molar mass of PCl₅ since it is the only component present initially.

\[ M_{avg\_initial} = \frac{(2n \text{ moles}) \times (M_{PCl_5})}{2n} = M_{PCl_5} = 208.24 \text{ g/mol} \]

### Step 3: Determine Moles After Decomposition
When PCl₅ undergoes 50% decomposition, it means that 50% of the initial moles of PCl₅ will decompose:

\[ \text{Moles of PCl}_5 \text{ remaining} = 2n - n = n \text{ moles} \]

### Step 4: Calculate Moles of Products Formed
From the decomposition reaction:

\[ PCl_5 \rightarrow PCl_3 + Cl_2 \]

For every mole of PCl₅ that decomposes, 1 mole of PCl₃ and 1 mole of Cl₂ are produced. Since 1 mole of PCl₅ decomposes (50% of 2n moles), we will have:

\[ \text{Moles of PCl}_3 = n \text{ moles} \]
\[ \text{Moles of Cl}_2 = n \text{ moles} \]

### Step 5: Calculate Total Moles After Decomposition
Now, the total moles in the mixture after decomposition will be:

\[ \text{Total moles} = \text{Moles of PCl}_5 + \text{Moles of PCl}_3 + \text{Moles of Cl}_2 = n + n + n = 3n \]

### Step 6: Calculate Total Mass After Decomposition
Now, we need to calculate the total mass of the mixture after decomposition:

\[ \text{Mass of PCl}_5 = n \times M_{PCl_5} = n \times 208.24 \text{ g/mol} \]
\[ \text{Mass of PCl}_3 = n \times M_{PCl_3} \quad (M_{PCl_3} \approx 137.33 \text{ g/mol}) \]
\[ \text{Mass of Cl}_2 = n \times M_{Cl_2} \quad (M_{Cl_2} \approx 70.90 \text{ g/mol}) \]

Calculating the total mass:

\[ \text{Total mass} = n \times 208.24 + n \times 137.33 + n \times 70.90 \]
\[ = n \times (208.24 + 137.33 + 70.90) = n \times 416.47 \text{ g} \]

### Step 7: Calculate Average Molar Mass After Decomposition (M_avg_final)
Now we can find the new average molar mass after decomposition:

\[ M_{avg\_final} = \frac{\text{Total mass}}{\text{Total moles}} = \frac{n \times 416.47}{3n} = \frac{416.47}{3} \approx 138.82 \text{ g/mol} \]

### Step 8: Calculate Percentage Change in Molar Mass
Now we can calculate the percentage change in average molar mass:

\[ \text{Percentage Change} = \frac{M_{avg\_final} - M_{avg\_initial}}{M_{avg\_initial}} \times 100 \]
\[ = \frac{138.82 - 208.24}{208.24} \times 100 \approx -33.33\% \]

### Final Answer
The percentage change in M_avg of the mixture after 50% decomposition of PCl₅ is approximately -33.33%.