What are the signs of the entropy change `( +` or `-)` in the following`:`
`I.` A liquird crystallisation in to solid
`II.` Temperature of solid raised from 0 to 115K
`III.` `2 NaHCO_(3)(s) rarr Na_(2)CO_(3)(s)+CO_(2)(g)+H_(2)O(g)`
`IV.` `H_(2)(g) rarr 2H(g)`

Options

A. `underset(-)(I)" " underset(+)(II)" "underset(+)(III)" "underset(+)(IV)`
B. `underset(-)(I)" " underset(-)(II)" "underset(+)(III)" "underset(+)(IV)`
C. `underset(-)(I)" " underset(-)(II)" "underset(-)(III)" "underset(+)(IV)`
D. `underset(+)(I)" " underset(-)(II)" "underset(-)(III)" "underset(-)(IV)`

Answer

To determine the signs of the entropy change in the given scenarios, we will analyze each case step by step.

### Step 1: Analyzing Liquid Crystallization to Solid
- When a liquid crystallizes into a solid, the molecules become more ordered and closely packed.
- In terms of entropy, which measures the disorder or randomness of a system, a solid has lower entropy than a liquid.
- Therefore, the entropy change for this process is negative.

**Entropy Change: Negative (-)**

### Step 2: Raising the Temperature of a Solid from 0 to 115 K
- When the temperature of a solid is increased, the kinetic energy of the molecules increases, causing them to vibrate more vigorously.
- This increase in molecular motion leads to a higher degree of disorder or randomness.
- Thus, the entropy of the solid increases with the increase in temperature.

**Entropy Change: Positive (+)**

### Step 3: Reaction of Sodium Bicarbonate
- The reaction given is:
\[ 2 \text{NaHCO}_3(s) \rightarrow \text{Na}_2\text{CO}_3(s) + \text{CO}_2(g) + \text{H}_2\text{O}(g) \]
- In this reaction, we start with 2 moles of solid sodium bicarbonate and produce 1 mole of solid sodium carbonate, 1 mole of gaseous carbon dioxide, and 1 mole of gaseous water.
- The total number of gaseous moles produced is 2 (from CO2 and H2O), while there are no gaseous reactants.
- The increase in the number of gaseous molecules indicates an increase in entropy.

**Entropy Change: Positive (+)**

### Step 4: Dissociation of Hydrogen Molecule
- The reaction is:
\[ \text{H}_2(g) \rightarrow 2 \text{H}(g) \]
- Here, one molecule of hydrogen gas dissociates into two hydrogen atoms.
- The number of particles has increased from 1 to 2, which indicates an increase in disorder.
- Therefore, the entropy change is positive.

**Entropy Change: Positive (+)**

### Summary of Entropy Changes
1. Liquid crystallizing to solid: **Negative (-)**
2. Temperature of solid raised from 0 to 115 K: **Positive (+)**
3. Reaction of sodium bicarbonate: **Positive (+)**
4. Dissociation of hydrogen molecule: **Positive (+)**

### Final Result
- The signs of the entropy change for the four cases are:
- I: (-)
- II: (+)
- III: (+)
- IV: (+)

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