In a reaction between `H_(2)` and `I_(2)` at a certain temperature, the amounts of `H_(2), I_(2)` and HI at equilibrium were found to be `0.45` mol, `0.39` mol, and `3.0` mol respectively. Calculate the equilibrium constant for the reaction at the given temperature.

Explanation

To calculate the equilibrium constant \( K_c \) for the reaction between \( H_2 \) and \( I_2 \) to form \( HI \), we will follow these steps:

### Step 1: Write the balanced chemical equation
The balanced chemical equation for the reaction is:
\[ H_2(g) + I_2(g) \rightleftharpoons 2 HI(g) \]

### Step 2: Identify the equilibrium concentrations
From the problem, we have the following equilibrium concentrations:
- Concentration of \( H_2 \) = 0.45 mol
- Concentration of \( I_2 \) = 0.39 mol
- Concentration of \( HI \) = 3.0 mol

### Step 3: Write the expression for the equilibrium constant \( K_c \)
The expression for the equilibrium constant \( K_c \) for the reaction is given by:
\[ K_c = \frac{[HI]^2}{[H_2][I_2]} \]
where \([HI]\), \([H_2]\), and \([I_2]\) are the molar concentrations of the respective species at equilibrium.

### Step 4: Substitute the equilibrium concentrations into the expression
Substituting the values into the \( K_c \) expression:
\[ K_c = \frac{(3.0)^2}{(0.45)(0.39)} \]

### Step 5: Calculate the numerator and denominator
Calculating the numerator:
\[ (3.0)^2 = 9.0 \]

Calculating the denominator:
\[ (0.45)(0.39) = 0.1755 \]

### Step 6: Calculate \( K_c \)
Now, substituting the calculated values into the equation:
\[ K_c = \frac{9.0}{0.1755} \approx 51.28 \]

### Step 7: State the final answer
Thus, the equilibrium constant \( K_c \) for the reaction at the given temperature is:
\[ K_c \approx 51.28 \]