For the reaction `A(g) iff B (g), K_(C)=10`
`B(g) iff C(g), K_(C)=2`
`C(g) iff D(g), K_(C)=0.01`
Calculate `K_(C)` for the reaction `D(g) iff A(g)`.

Answer

To solve the problem of calculating \( K_c \) for the reaction \( D(g) \iff A(g) \) given the equilibrium constants for three other reactions, we can follow these steps:

### Step 1: Write down the given reactions and their equilibrium constants.
1. \( A(g) \iff B(g) \) with \( K_{c1} = 10 \)
2. \( B(g) \iff C(g) \) with \( K_{c2} = 2 \)
3. \( C(g) \iff D(g) \) with \( K_{c3} = 0.01 \)

### Step 2: Combine the reactions.
To find the equilibrium constant for \( D(g) \iff A(g) \), we can add the three reactions together. The addition of the reactions can be represented as follows:

- From \( A \iff B \) (1)
- From \( B \iff C \) (2)
- From \( C \iff D \) (3)

When we add these reactions, we get:
\[ A(g) + B(g) + C(g) \iff B(g) + C(g) + D(g) \]

### Step 3: Cancel out the common species.
In the combined reaction, \( B \) and \( C \) appear on both sides and can be canceled out:
\[ A(g) \iff D(g) \]

### Step 4: Write the equilibrium constant for the combined reaction.
The equilibrium constant for the combined reaction \( A(g) \iff D(g) \) can be calculated by multiplying the equilibrium constants of the individual reactions:
\[ K_{c4} = K_{c1} \times K_{c2} \times K_{c3} \]

### Step 5: Substitute the values of the equilibrium constants.
Now substitute the values:
\[ K_{c4} = 10 \times 2 \times 0.01 \]

### Step 6: Calculate \( K_{c4} \).
Calculating this gives:
\[ K_{c4} = 10 \times 2 \times 0.01 = 0.2 \]

### Step 7: Find the equilibrium constant for the reverse reaction.
Since we need \( K_c' \) for the reaction \( D(g) \iff A(g) \), we take the reciprocal of \( K_{c4} \):
\[ K_c' = \frac{1}{K_{c4}} = \frac{1}{0.2} = 5 \]

### Final Answer:
Thus, the equilibrium constant \( K_c \) for the reaction \( D(g) \iff A(g) \) is:
\[ K_c' = 5 \]

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