At 500K, equilibrium constant, `K_c` for the following reaction is 5.
`1/(2) H_2(g) + 1/2I_(2)(g) hArr HI(g)`
What would be the equilibrium constant K. for the reaction `2HI(g) hArr H_2(g) +I_2(g)`

Explanation

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

\[ 2 \text{HI}(g) \rightleftharpoons \text{H}_2(g) + \text{I}_2(g) \]

given that the equilibrium constant \( K_c \) for the reaction

\[ \frac{1}{2} \text{H}_2(g) + \frac{1}{2} \text{I}_2(g) \rightleftharpoons \text{HI}(g) \]

is 5 at 500 K, we can follow these steps:

### Step 1: Write the expression for \( K_c \) for the first reaction.

For the reaction

\[ \frac{1}{2} \text{H}_2(g) + \frac{1}{2} \text{I}_2(g) \rightleftharpoons \text{HI}(g) \]

the equilibrium constant \( K_c \) is given by:

\[ K_c = \frac{[\text{HI}]}{[\text{H}_2]^{1/2} [\text{I}_2]^{1/2}} \]

Given that \( K_c = 5 \), we have:

\[ 5 = \frac{[\text{HI}]}{[\text{H}_2]^{1/2} [\text{I}_2]^{1/2}} \]

### Step 2: Write the expression for \( K_c' \) for the second reaction.

For the reaction

\[ 2 \text{HI}(g) \rightleftharpoons \text{H}_2(g) + \text{I}_2(g) \]

the equilibrium constant \( K_c' \) is given by:

\[ K_c' = \frac{[\text{H}_2][\text{I}_2]}{[\text{HI}]^2} \]

### Step 3: Relate \( K_c' \) to \( K_c \).

Since the second reaction is the reverse of the first reaction and the coefficients are multiplied by 2, we can relate \( K_c' \) to \( K_c \) as follows:

1. When a reaction is reversed, the equilibrium constant is the reciprocal of the original constant:

\[ K_c' = \frac{1}{K_c} \]

2. When the coefficients of a balanced equation are multiplied by a factor, the equilibrium constant is raised to the power of that factor. In this case, since we are doubling the coefficients (2 in front of HI), we square the equilibrium constant:

\[ K_c' = \frac{1}{K_c^2} \]

### Step 4: Substitute the value of \( K_c \).

Now, substituting \( K_c = 5 \):

\[ K_c' = \frac{1}{5^2} = \frac{1}{25} \]

### Step 5: Calculate \( K_c' \).

Calculating \( K_c' \):

\[ K_c' = 0.04 \]

### Final Answer:

The equilibrium constant \( K_c' \) for the reaction

\[ 2 \text{HI}(g) \rightleftharpoons \text{H}_2(g) + \text{I}_2(g) \]

is \( 0.04 \).

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