`S(s)+O_(2)hArrSO_(2)(g)" "K_(1)=10^(52)`
`2S(s)+3O_(2)hArr2SO_(3)(g)" "K_(2)=10^(129)`
Calculate `K_("equilibrium")` for

Options

A. `10^(25)`
B. `10^(77)`
C. `10^(70)`
D. `10^(40)`

Answer

To calculate the equilibrium constant \( K_{\text{equilibrium}} \) for the reaction:

\[ 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \]

we will use the provided 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 reactions and their equilibrium constants

We have:

- Reaction 1:
\[ S(s) + O_2(g) \rightleftharpoons SO_2(g) \quad K_1 = 10^{52} \]

- Reaction 2:
\[ 2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g) \quad K_2 = 10^{129} \]

### Step 2: Modify Reaction 1

To use Reaction 1 for our target reaction, we need to multiply it by 2:

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

The equilibrium constant for this modified reaction, \( K_1' \), is given by:

\[ K_1' = K_1^2 = (10^{52})^2 = 10^{104} \]

### Step 3: Write the modified Reaction 1 and Reaction 2

Now we have:

- Modified Reaction 1:
\[ 2S(s) + 2O_2(g) \rightleftharpoons 2SO_2(g) \quad K_1' = 10^{104} \]

- Reaction 2 remains:
\[ 2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g) \quad K_2 = 10^{129} \]

### Step 4: Combine the reactions

To find the equilibrium constant for the target reaction, we subtract the modified Reaction 1 from Reaction 2:

\[ \begin{align*} (2S(s) + 3O_2(g) \rightleftharpoons 2SO_3(g)) \\ - (2S(s) + 2O_2(g) \rightleftharpoons 2SO_2(g)) \end{align*} \]

This gives us:

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

### Step 5: Write the new equilibrium expression

The new equilibrium constant \( K_3 \) for the reaction:

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

is given by:

\[ K_3 = \frac{K_2}{K_1'} = \frac{10^{129}}{10^{104}} = 10^{25} \]

### Final Answer

Thus, the equilibrium constant \( K_{\text{equilibrium}} \) for the reaction \( 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \) is:

\[ K_{\text{equilibrium}} = 10^{25} \]