The equilibrium `SO_(2)(g)+(1)/(2)O_(2)(g)hArrSO_(3)(g)` is established in a container of `4L` at a particular temperature. If the number of moles of `SO_(2),O_(2) "and" SO_(3)` at equilibrium are `2,1 "and" 4` respectively then find the value of equilibrium constant.

Explanation

To find the equilibrium constant \( K_c \) for the reaction

\[ SO_2(g) + \frac{1}{2} O_2(g) \rightleftharpoons SO_3(g) \]

given the number of moles at equilibrium, we can follow these steps:

### Step 1: Write down the equilibrium expression
The equilibrium constant \( K_c \) is defined as:

\[ K_c = \frac{[SO_3]}{[SO_2][O_2]^{1/2}} \]

where \([SO_3]\), \([SO_2]\), and \([O_2]\) are the molar concentrations of the respective gases at equilibrium.

### Step 2: Calculate the concentrations of each species
We know the number of moles of each gas at equilibrium and the volume of the container (4 L). The concentration can be calculated using the formula:

\[ \text{Concentration} = \frac{\text{Number of moles}}{\text{Volume (L)}} \]

- For \( SO_2 \):
\[ [SO_2] = \frac{2 \text{ moles}}{4 \text{ L}} = 0.5 \, \text{M} \]

- For \( O_2 \):
\[ [O_2] = \frac{1 \text{ mole}}{4 \text{ L}} = 0.25 \, \text{M} \]

- For \( SO_3 \):
\[ [SO_3] = \frac{4 \text{ moles}}{4 \text{ L}} = 1 \, \text{M} \]

### Step 3: Substitute the concentrations into the equilibrium expression
Now we can substitute these concentrations into the equilibrium expression:

\[ K_c = \frac{[SO_3]}{[SO_2][O_2]^{1/2}} = \frac{1}{(0.5)(0.25)^{1/2}} \]

### Step 4: Calculate \( K_c \)
Calculating \( (0.25)^{1/2} \):

\[ (0.25)^{1/2} = 0.5 \]

Now substituting this back into the equation:

\[ K_c = \frac{1}{(0.5)(0.5)} = \frac{1}{0.25} = 4 \]

### Final Answer
Thus, the value of the equilibrium constant \( K_c \) is:

\[ K_c = 4 \]

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