For the gas phase reaction
`PCl_(5)rarrPCl_(3)(g)+Cl_(2)(g)`
which of the following conditions are correct?

Options

A. `DeltaH=0 and DeltaS lt 0`
B. `DeltaH gt 0 and DeltaS gt 0`
C. `DeltaH lt 0 and DeltaS lt 0`
D. `DeltaH gt 0 and DeltaS lt 0`

Explanation

To solve the question regarding the gas phase reaction \( PCl_5 \rightarrow PCl_3 (g) + Cl_2 (g) \), we need to analyze the changes in enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)) for the reaction. Here’s a step-by-step breakdown of the solution:

### Step 1: Write the Reaction
The reaction is given as:
\[ PCl_5 (g) \rightarrow PCl_3 (g) + Cl_2 (g) \]

### Step 2: Calculate \( \Delta N_G \)
To find \( \Delta N_G \), we need to determine the change in the number of moles of gas during the reaction.

- **Reactants**: 1 mole of \( PCl_5 \)
- **Products**: 1 mole of \( PCl_3 \) + 1 mole of \( Cl_2 \) = 2 moles

Now, we can calculate \( \Delta N_G \):
\[ \Delta N_G = \text{(moles of products)} - \text{(moles of reactants)} = 2 - 1 = 1 \]

### Step 3: Analyze Change in Enthalpy (\( \Delta H \))
Using the relationship between enthalpy change and internal energy change:
\[ \Delta H = \Delta E + \Delta N_G RT \]
Since \( \Delta N_G \) is greater than zero (1), it implies that \( \Delta H \) is also greater than zero:
\[ \Delta H > 0 \]

### Step 4: Analyze Change in Entropy (\( \Delta S \))
For the change in entropy, we can use the general idea that the entropy of a system increases when there is an increase in the number of gas molecules. Here, we have gone from 1 mole of \( PCl_5 \) to 2 moles of products, indicating an increase in disorder:
\[ \Delta S > 0 \]

### Step 5: Conclusion
From the analysis, we have determined:
- \( \Delta H > 0 \)
- \( \Delta S > 0 \)

### Step 6: Identify the Correct Condition
Now, we need to check the options provided in the question:
- The correct option must state that both \( \Delta H \) and \( \Delta S \) are greater than zero.

### Final Answer
The correct condition for the reaction is:
- \( \Delta H > 0 \) and \( \Delta S > 0 \)