At a certain temperature and a total pressure of `10^(5) Pa`, iodine vapour contains `40%` by volume of `I "atoms"`, Calculate `K_(p)` for the equilibrium.
`I_(2(g))hArr2I_((g))`

Explanation

To solve the problem of calculating \( K_p \) for the equilibrium \( I_2(g) \rightleftharpoons 2I(g) \) at a total pressure of \( 10^5 \) Pa, where iodine vapor contains 40% by volume of iodine atoms, we can follow these steps:

### Step 1: Identify the given data
- Total pressure, \( P_{total} = 10^5 \) Pa
- Volume percentage of iodine atoms, \( V_{I} = 40\% \)

### Step 2: Calculate the mole fractions
Assuming a total of 100 moles of gas:
- Moles of iodine atoms, \( n_I = 40 \)
- Moles of \( I_2 \), \( n_{I_2} = 100 - 40 = 60 \)

Now, we can calculate the mole fractions:
- Mole fraction of iodine atoms, \( \chi_I = \frac{n_I}{n_{total}} = \frac{40}{100} = 0.4 \)
- Mole fraction of \( I_2 \), \( \chi_{I_2} = \frac{n_{I_2}}{n_{total}} = \frac{60}{100} = 0.6 \)

### Step 3: Calculate the partial pressures
Using Dalton's law of partial pressures:
- Partial pressure of iodine atoms, \( P_I = P_{total} \times \chi_I = 10^5 \times 0.4 = 4 \times 10^4 \) Pa
- Partial pressure of \( I_2 \), \( P_{I_2} = P_{total} \times \chi_{I_2} = 10^5 \times 0.6 = 6 \times 10^4 \) Pa

### Step 4: Write the expression for \( K_p \)
The equilibrium constant \( K_p \) for the reaction \( I_2(g) \rightleftharpoons 2I(g) \) is given by:
\[ K_p = \frac{(P_I)^2}{P_{I_2}} \]

### Step 5: Substitute the values into the \( K_p \) expression
Substituting the calculated partial pressures into the expression:
\[ K_p = \frac{(4 \times 10^4)^2}{6 \times 10^4} \]

### Step 6: Simplify the expression
Calculating \( (4 \times 10^4)^2 \):
\[ (4 \times 10^4)^2 = 16 \times 10^8 \]
Now substituting this back into the equation:
\[ K_p = \frac{16 \times 10^8}{6 \times 10^4} = \frac{16}{6} \times 10^{8-4} = \frac{8}{3} \times 10^4 \]

### Step 7: Final calculation
Calculating \( \frac{8}{3} \):
\[ K_p \approx 2.67 \times 10^4 \text{ Pa} \]

### Final Answer
Thus, the value of \( K_p \) is approximately \( 2.67 \times 10^4 \) Pa.

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