Phosgene, `COCl_(2)`, a poisonous gas decomposes according to the equation
`COCl_(2)(g)hArrCO(g)+Cl_(2)(g)`
If `K_(c)=0.083` at `900^(@)C`, What is the value of `K_(p)`?

Options

A. 0.125
B. `8.0`
C. `6.1`
D. `0.16`

Answer

To find the value of \( K_p \) for the decomposition of phosgene \( COCl_2 \) according to the equation:

\[ COCl_2(g) \rightleftharpoons CO(g) + Cl_2(g) \]

we can use the relationship between \( K_p \) and \( K_c \):

\[ K_p = K_c \cdot R^{\Delta N_g} \cdot T \]

Where:
- \( K_c \) is the equilibrium constant in terms of concentration.
- \( R \) is the universal gas constant.
- \( T \) is the temperature in Kelvin.
- \( \Delta N_g \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

### Step 1: Identify the values
Given:
- \( K_c = 0.083 \)
- Temperature \( T = 900^\circ C \)

### Step 2: Convert temperature to Kelvin
To convert Celsius to Kelvin, we use the formula:

\[ T(K) = T(°C) + 273 \]

So,

\[ T = 900 + 273 = 1173 \, K \]

### Step 3: Calculate \( \Delta N_g \)
Next, we calculate \( \Delta N_g \):

\[ \Delta N_g = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \]

From the balanced equation, we have:
- Products: 1 mole of \( CO \) and 1 mole of \( Cl_2 \) (total 2 moles)
- Reactants: 1 mole of \( COCl_2 \)

Thus,

\[ \Delta N_g = 2 - 1 = 1 \]

### Step 4: Use the gas constant
The value of the gas constant \( R \) is:

\[ R = 0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \]

### Step 5: Substitute values into the equation for \( K_p \)
Now we can substitute the values into the equation for \( K_p \):

\[ K_p = K_c \cdot R^{\Delta N_g} \cdot T \]

Substituting the known values:

\[ K_p = 0.083 \cdot (0.082)^{1} \cdot (1173) \]

### Step 6: Calculate \( K_p \)
Calculating \( K_p \):

\[ K_p = 0.083 \cdot 0.082 \cdot 1173 \]

Calculating the product:

\[ K_p = 0.083 \cdot 0.082 = 0.006806 \]

Then,

\[ K_p = 0.006806 \cdot 1173 \approx 7.98 \]

### Step 7: Round the result
Rounding \( 7.98 \) gives us:

\[ K_p \approx 8 \]

### Final Answer:
Thus, the value of \( K_p \) is approximately \( 8 \).

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