the incorrect expression among the following is

Options

A. `K=e^(−ΔG^(@)/R T)`
B. In `K =( DeltaH° - TDeltaS°)/RT`
C. `DeltaS_sum = -DeltaH_sys /T`
D. `DeltaS_sys = q_rev/T`

Answer

To find the incorrect expression among the given options, we will analyze each expression step by step.

### Step 1: Analyze the first expression
The first expression is:
\[ k = e^{-\frac{\Delta G^\circ}{RT}} \]
This expression relates the equilibrium constant \( k \) to the standard Gibbs free energy change \( \Delta G^\circ \).

**Verification:**
From thermodynamics, we know:
\[ \Delta G^\circ = -RT \ln k \]
Rearranging gives:
\[ \ln k = -\frac{\Delta G^\circ}{RT} \]
Exponentiating both sides results in:
\[ k = e^{-\frac{\Delta G^\circ}{RT}} \]
Thus, this expression is **correct**.

### Step 2: Analyze the second expression
The second expression is:
\[ \ln k = \frac{\Delta H^\circ - T \Delta S^\circ}{RT} \]

**Verification:**
From the Gibbs free energy equation:
\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]
And knowing that:
\[ \Delta G^\circ = -RT \ln k \]
We can substitute:
\[ -RT \ln k = \Delta H^\circ - T \Delta S^\circ \]
Rearranging gives:
\[ \ln k = \frac{\Delta H^\circ - T \Delta S^\circ}{RT} \]
This expression is **correct**.

### Step 3: Analyze the third expression
The third expression is:
\[ \Delta S_{\text{total}} = -\frac{\Delta H_{\text{system}}}{T} \]

**Verification:**
In an exothermic reaction, the total entropy change can be expressed as:
\[ \Delta S_{\text{total}} = \Delta S_{\text{surroundings}} + \Delta S_{\text{system}} \]
For exothermic reactions, we have:
\[ \Delta S_{\text{surroundings}} = \frac{\Delta H_{\text{system}}}{T} \]
Thus, the total entropy change becomes:
\[ \Delta S_{\text{total}} = \frac{\Delta H_{\text{system}}}{T} + \Delta S_{\text{system}} \]
This expression is **correct**.

### Step 4: Analyze the fourth expression
The fourth expression is:
\[ \Delta S_{\text{system}} = \frac{Q_{\text{rev}}}{T} \]

**Verification:**
For a reversible process at constant temperature, the change in entropy is given by:
\[ \Delta S = \frac{Q_{\text{rev}}}{T} \]
This expression is **correct**.

### Conclusion
After analyzing all the expressions, we find that the second expression:
\[ \ln k = \frac{\Delta H^\circ - T \Delta S^\circ}{RT} \]
is incorrect. The correct expression should be:
\[ \ln k = \frac{\Delta S^\circ - \Delta H^\circ}{RT} \]

### Final Answer
The incorrect expression among the following is:
**Option B: \(\ln k = \frac{\Delta H^\circ - T \Delta S^\circ}{RT}\)**

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