In each of the following, total pressure set - up at equilibrium is assumed to be equal and is P atm with equilibrium constants `K_(p)` given `:`
`I: CaCO_(3)(s)hArr CaO(s)+CO_(2)(g),K_(1)`
`II : NH_(4)HS(s)hArr NH_(3)(g)+H_(2)S(g),K_(2)`
`III : NH_(2)CO_(2)NH_(4)(s)hArr 2NH_(3)(g) +CO_(2)(g),K_(3)`
In the increasing order `:`

Options

A. `K_(1)=K_(2)=K_(3)`
B. `K_(1) lt K_(2) lt K_(3)`
C. `K_(3)lt K_(2) lt K_(1)`
D. None of these

Answer

To determine the increasing order of the equilibrium constants \( K_1, K_2, \) and \( K_3 \) for the given reactions, we will analyze each reaction step by step.

### Step 1: Analyze the first reaction
**Reaction I:**
\[ \text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g) \]

- The only gaseous product is \( \text{CO}_2 \).
- Since the total pressure at equilibrium is \( P \) atm, the pressure of \( \text{CO}_2 \) will be equal to \( P \).
- Therefore, the equilibrium constant \( K_1 \) is given by:
\[ K_1 = P_{CO_2} = P \]

### Step 2: Analyze the second reaction
**Reaction II:**
\[ \text{NH}_4\text{HS}(s) \rightleftharpoons \text{NH}_3(g) + \text{H}_2\text{S}(g) \]

- There are two gaseous products: \( \text{NH}_3 \) and \( \text{H}_2\text{S} \).
- To maintain the total pressure \( P \), we can assume that the pressures of \( \text{NH}_3 \) and \( \text{H}_2\text{S} \) are equal due to their stoichiometric coefficients being 1.
- Thus, the pressure of each gas will be \( \frac{P}{2} \).
- Therefore, the equilibrium constant \( K_2 \) is given by:
\[ K_2 = P_{NH_3} \cdot P_{H_2S} = \left(\frac{P}{2}\right) \cdot \left(\frac{P}{2}\right) = \frac{P^2}{4} \]

### Step 3: Analyze the third reaction
**Reaction III:**
\[ \text{NH}_2\text{CO}_2\text{NH}_4(s) \rightleftharpoons 2\text{NH}_3(g) + \text{CO}_2(g) \]

- Here, we have two moles of \( \text{NH}_3 \) and one mole of \( \text{CO}_2 \), making a total of 3 gaseous moles.
- The total pressure is still \( P \). To find the individual pressures:
- Let \( P_{NH_3} = \frac{2P}{3} \) (since there are 2 moles of \( \text{NH}_3 \))
- Let \( P_{CO_2} = \frac{P}{3} \)
- Therefore, the equilibrium constant \( K_3 \) is given by:
\[ K_3 = (P_{NH_3})^2 \cdot P_{CO_2} = \left(\frac{2P}{3}\right)^2 \cdot \left(\frac{P}{3}\right) = \frac{4P^2}{9} \cdot \frac{P}{3} = \frac{4P^3}{27} \]

### Step 4: Compare the values of \( K_1, K_2, \) and \( K_3 \)
- We have:
- \( K_1 = P \)
- \( K_2 = \frac{P^2}{4} \)
- \( K_3 = \frac{4P^3}{27} \)

### Step 5: Determine the increasing order
To compare these values, we can express them in terms of \( P \):

1. \( K_1 = P \)
2. \( K_2 = \frac{P^2}{4} \)
3. \( K_3 = \frac{4P^3}{27} \)

Now, we can evaluate the relative sizes:
- Since \( P \) is a constant, we can see that:
- \( K_3 \) will be the smallest because it is a cubic term divided by a larger constant.
- \( K_2 \) is next because it is a quadratic term divided by 4.
- \( K_1 \) is the largest as it is linear.

Thus, the increasing order of the equilibrium constants is:
\[ K_3 < K_2 < K_1 \]

### Final Answer
The increasing order is: \( K_3 < K_2 < K_1 \).