Variation of equilibrium constan K with temperature is given by van't Hoff equation
`InK=(Delta_(r)S^(@))/R-(Delta_(r)H^(@))/(RT)`
for this equation, `(Delta_(r)H^(@))` can be evaluated if equilibrium constants `K_(1)` and `K_(2)` at two temperature `T_(1)` and `T_(2)` are known.
`log(K_(2)/K_(1))=(Delta_(r)H^(@))/(2.303R)[1/T_(1)-1/T_(2)]`
For an isomerization `X(g)hArrY(g)` the temperature dependency of equilibrium constant is given by :
`lnK=2-(1000)/T`
The value of `Delta_(r)S^(@)` at `300 K` is :

Options

A. `2` R
B. `2/R`
C. `1000` R
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given equation
The temperature dependency of the equilibrium constant \( K \) is given by:
\[ \ln K = 2 - \frac{1000}{T} \]
This equation can be used to find the change in standard entropy \( \Delta_r S^\circ \) at a temperature of 300 K.

### Step 2: Differentiate the equation
To find the change in standard entropy \( \Delta_r S^\circ \), we can differentiate the equation with respect to temperature \( T \):
\[ \frac{d(\ln K)}{dT} = \frac{1000}{T^2} \]
From thermodynamic principles, we know that:
\[ \frac{d(\ln K)}{dT} = \frac{\Delta_r H^\circ}{RT^2} - \frac{\Delta_r S^\circ}{R} \]
Setting these two expressions equal gives us:
\[ \frac{\Delta_r H^\circ}{RT^2} - \frac{\Delta_r S^\circ}{R} = \frac{1000}{T^2} \]

### Step 3: Rearranging the equation
Multiplying through by \( R \) to eliminate the denominator:
\[ \Delta_r H^\circ \frac{1}{T^2} - \Delta_r S^\circ = 1000 \]
Rearranging gives:
\[ \Delta_r S^\circ = \Delta_r H^\circ \frac{1}{T^2} - 1000 \]

### Step 4: Substitute \( T = 300 \, K \)
Now we can substitute \( T = 300 \, K \) into the equation. We need to find \( \Delta_r H^\circ \) first.

### Step 5: Finding \( \Delta_r H^\circ \)
From the equation \( \ln K = 2 - \frac{1000}{T} \), we can find \( K \) at \( T = 300 \, K \):
\[ \ln K = 2 - \frac{1000}{300} = 2 - 3.33 = -1.33 \]
Thus:
\[ K = e^{-1.33} \approx 0.26 \]

### Step 6: Calculate \( \Delta_r H^\circ \)
Using the relationship of \( K \) at two temperatures (though we have only one temperature here), we can assume that the enthalpy change \( \Delta_r H^\circ \) can be approximated from the slope of the \( \ln K \) vs \( \frac{1}{T} \) graph.

### Step 7: Find \( \Delta_r S^\circ \)
Using the value of \( \Delta_r H^\circ \) (which we can approximate from the temperature dependency), we can substitute back into the equation for \( \Delta_r S^\circ \):
\[ \Delta_r S^\circ = 2R \]

### Final Answer
Thus, the value of \( \Delta_r S^\circ \) at 300 K is:
\[ \Delta_r S^\circ = 2R \]