In how many of the following entropy increases?
`{:((a) N_(2(g)) + 3H_(2(g)) rarr 2NH_(3(g)),(b)PCl_(5(g)) rarr PCl_(3(g)) + Cl_(2(g))),((c ) H_(2)O_((s)) rarr H_(2)O_((g)),(d)H_(2)O_((l)) rarr H_(2)O_((g))),((e ) 2NaHCO_(3(s)) rarr Na_(2)CO_(3(s)) + H_(2)O_((g)) + CO_(2(g)),(f ) NH_(2)CO_(2)NH_(4(s)) rarr 2NH_(3(g)) + CO_(2(g))),((g) H_(2)O_((l)) rarr H_(2)O_((s)),(h) CO_(2(s)) rarr CO_(2(g))):}`

Answer

To determine in how many of the given reactions the entropy increases, we will analyze each reaction step by step. Entropy is a measure of disorder in a system, and generally, gases have higher entropy than liquids, which in turn have higher entropy than solids.

Let's evaluate each option:

### Step-by-Step Solution:

1. **Reaction (a):**
\[ N_2(g) + 3H_2(g) \rightarrow 2NH_3(g) \]
- **Reactants:** 4 moles of gas (1 mole of \(N_2\) + 3 moles of \(H_2\))
- **Products:** 2 moles of gas (\(2NH_3\))
- **Entropy Change:** Decreases (4 moles to 2 moles).

2. **Reaction (b):**
\[ PCl_5(g) \rightarrow PCl_3(g) + Cl_2(g) \]
- **Reactants:** 1 mole of gas (\(PCl_5\))
- **Products:** 2 moles of gas (\(PCl_3 + Cl_2\))
- **Entropy Change:** Increases (1 mole to 2 moles).

3. **Reaction (c):**
\[ H_2O(s) \rightarrow H_2O(g) \]
- **Reactants:** 1 mole of solid
- **Products:** 1 mole of gas
- **Entropy Change:** Increases (solid to gas).

4. **Reaction (d):**
\[ H_2O(l) \rightarrow H_2O(g) \]
- **Reactants:** 1 mole of liquid
- **Products:** 1 mole of gas
- **Entropy Change:** Increases (liquid to gas).

5. **Reaction (e):**
\[ 2NaHCO_3(s) \rightarrow Na_2CO_3(s) + H_2O(g) + CO_2(g) \]
- **Reactants:** 2 moles of solid
- **Products:** 1 mole of solid + 2 moles of gas
- **Entropy Change:** Increases (2 moles of solid to 1 mole of solid + 2 moles of gas).

6. **Reaction (f):**
\[ NH_2CO_2NH_4(s) \rightarrow 2NH_3(g) + CO_2(g) \]
- **Reactants:** 1 mole of solid
- **Products:** 2 moles of gas + 1 mole of gas
- **Entropy Change:** Increases (solid to gas).

7. **Reaction (g):**
\[ H_2O(l) \rightarrow H_2O(s) \]
- **Reactants:** 1 mole of liquid
- **Products:** 1 mole of solid
- **Entropy Change:** Decreases (liquid to solid).

8. **Reaction (h):**
\[ CO_2(s) \rightarrow CO_2(g) \]
- **Reactants:** 1 mole of solid
- **Products:** 1 mole of gas
- **Entropy Change:** Increases (solid to gas).

### Summary of Entropy Changes:
- **Increased Entropy:** (b), (c), (d), (e), (f), (h)
- **Decreased Entropy:** (a), (g)

### Total Count of Reactions with Increased Entropy:
There are **6 reactions** where entropy increases.

### Final Answer:
**6 reactions** show an increase in entropy.

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