For the following reactions , equilibrium constants are given :
`S(s)+O_2(g) hArr SO_2(g) , K_1=10^52`
`2S(s) +3O_2(g) hArr 2SO_3(g) , K_2=10^129`
The equilibrium constant for the reaction ,
`2SO_2(g) +O_2(g) hArr 2SO_3(g)` is :

Options

A. `10^181`
B. `10^154`
C. `10^25`
D. `10^77`

Answer

To find the equilibrium constant for the reaction

\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

we will use the given equilibrium constants \( K_1 \) and \( K_2 \) for the following reactions:

1. \( S(s) + O_2(g) \rightleftharpoons SO_2(g) \), with \( K_1 = 10^{52} \)
2. \( 2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g) \), with \( K_2 = 10^{129} \)

### Step 1: Write the reaction for \( SO_2 \)

We need to express \( SO_2 \) in terms of \( S \) and \( O_2 \). The first reaction gives us \( SO_2 \):

\[ S(s) + O_2(g) \rightleftharpoons SO_2(g) \]

### Step 2: Reverse the first reaction

To obtain \( 2SO_2(g) \), we need to reverse the first reaction and multiply it by 2:

\[ 2SO_2(g) \rightleftharpoons 2S(s) + 2O_2(g) \]

The equilibrium constant for the reversed reaction is:

\[ K' = \frac{1}{K_1^2} = \frac{1}{(10^{52})^2} = 10^{-104} \]

### Step 3: Use the second reaction

Now, we can use the second reaction as it is:

\[ 2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g) \]

with \( K_2 = 10^{129} \).

### Step 4: Add the two reactions

Now we can add the modified first reaction and the second reaction:

1. \( 2SO_2(g) \rightleftharpoons 2S(s) + 2O_2(g) \) (with \( K' = 10^{-104} \))
2. \( 2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g) \) (with \( K_2 = 10^{129} \))

When we add these two reactions, the \( 2S(s) \) cancels out:

\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

### Step 5: Calculate the overall equilibrium constant

The overall equilibrium constant \( K \) for the reaction is the product of the equilibrium constants of the individual reactions:

\[ K = K' \times K_2 \]

Substituting the values:

\[ K = 10^{-104} \times 10^{129} = 10^{-104 + 129} = 10^{25} \]

### Final Answer

Thus, the equilibrium constant for the reaction

\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

is

\[ K = 10^{25} \]

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