For reactions `A rarr B` and `P rarr Q` Arrhenius constant are `10^(8)` and `10^(10)` respectively. If `E_(A rarrB) = 600 cal //` mole and `E_(P rarr Q )= 1200 cal //` mole, then find the temperature at which their rate constants are same ( Given `: R =2 cal //` mole `//` K )

Options

A. 600K
B. `300xx 4.606 K`
C. `( 300)/( 4.606)K`
D. `( 4.606)/( 600)K`

Answer

To find the temperature at which the rate constants for the reactions \( A \rightarrow B \) and \( P \rightarrow Q \) are the same, we can use the Arrhenius equation:

\[ k = A e^{-\frac{E_a}{RT}} \]

where:
- \( k \) is the rate constant,
- \( A \) is the Arrhenius constant,
- \( E_a \) is the activation energy,
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin.

### Step 1: Write the Arrhenius equations for both reactions

For the reaction \( A \rightarrow B \):
\[ k_{A \rightarrow B} = 10^8 e^{-\frac{600}{2T}} \]

For the reaction \( P \rightarrow Q \):
\[ k_{P \rightarrow Q} = 10^{10} e^{-\frac{1200}{2T}} \]

### Step 2: Set the rate constants equal to each other

Since we want to find the temperature at which the rate constants are the same, we set the two equations equal to each other:

\[ 10^8 e^{-\frac{600}{2T}} = 10^{10} e^{-\frac{1200}{2T}} \]

### Step 3: Simplify the equation

Dividing both sides by \( 10^8 \):

\[ e^{-\frac{600}{2T}} = 10^2 e^{-\frac{1200}{2T}} \]

This simplifies to:

\[ e^{-\frac{600}{2T}} = 100 e^{-\frac{1200}{2T}} \]

### Step 4: Rearranging the equation

Taking the natural logarithm on both sides:

\[ -\frac{600}{2T} = \ln(100) - \frac{1200}{2T} \]

### Step 5: Solve for \( T \)

Rearranging gives:

\[ -\frac{600}{2T} + \frac{1200}{2T} = \ln(100) \]

This simplifies to:

\[ \frac{600}{2T} = \ln(100) \]

Now substituting \( \ln(100) = 2 \ln(10) \approx 4.605 \):

\[ \frac{600}{2T} = 4.605 \]

Multiplying both sides by \( 2T \):

\[ 600 = 4.605 \times 2T \]

Now solving for \( T \):

\[ T = \frac{600}{4.605 \times 2} \]

Calculating the right-hand side:

\[ T = \frac{600}{9.21} \approx 65.1 \text{ K} \]

### Conclusion

The temperature at which the rate constants for the two reactions are the same is approximately:

\[ T \approx \frac{300}{4.605} \text{ K} \]