P and Q arer two elements which form `P_(2)Q_(3)` and `PQ_(2)`. If 0.15 mole of `P_(2)Q_(3)` weights 15.9 g and 0.15 mole og `PQ_(2)` weighs 9.3 g then, what are atomic weights of P and Q ?

Answer

To solve the problem, we need to determine the atomic weights of elements P and Q based on the molecular weights of the compounds they form, \( P_2Q_3 \) and \( PQ_2 \).

### Step 1: Calculate the molecular weight of \( P_2Q_3 \)

Given:
- Weight of \( P_2Q_3 \) = 15.9 g
- Moles of \( P_2Q_3 \) = 0.15 moles

Using the formula for molecular weight:
\[ \text{Molecular Weight} = \frac{\text{Weight}}{\text{Number of Moles}} \]
\[ \text{Molecular Weight of } P_2Q_3 = \frac{15.9 \, \text{g}}{0.15 \, \text{moles}} = 106 \, \text{g/mol} \]

### Step 2: Calculate the molecular weight of \( PQ_2 \)

Given:
- Weight of \( PQ_2 \) = 9.3 g
- Moles of \( PQ_2 \) = 0.15 moles

Using the same formula for molecular weight:
\[ \text{Molecular Weight of } PQ_2 = \frac{9.3 \, \text{g}}{0.15 \, \text{moles}} = 62 \, \text{g/mol} \]

### Step 3: Set up equations based on molecular weights

From the molecular formulas, we can set up the following equations based on the molecular weights calculated:

1. For \( P_2Q_3 \):
\[ 2P + 3Q = 106 \quad \text{(Equation 1)} \]

2. For \( PQ_2 \):
\[ P + 2Q = 62 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations

We can solve these two equations simultaneously.

First, let's express \( P \) from Equation 2:
\[ P = 62 - 2Q \quad \text{(Equation 3)} \]

Now, substitute Equation 3 into Equation 1:
\[ 2(62 - 2Q) + 3Q = 106 \]
Expanding this:
\[ 124 - 4Q + 3Q = 106 \]
Combining like terms:
\[ 124 - Q = 106 \]
Solving for \( Q \):
\[ Q = 124 - 106 = 18 \]

### Step 5: Find \( P \)

Now substitute the value of \( Q \) back into Equation 3:
\[ P = 62 - 2(18) \]
\[ P = 62 - 36 = 26 \]

### Conclusion

The atomic weights of elements P and Q are:
- Atomic weight of P = 26 g/mol
- Atomic weight of Q = 18 g/mol