The decomposition of hydrogen peroxide follows the equation:

\(K = \left(2.9 \times 10^{14}s^{-1}\right)e^{\frac{-21000}{T}}\)

The value of \(E_a\) is:

174.594 kJ/mol

The correct answer is option 3. 174.594 kJ/mol.

The decomposition of hydrogen peroxide follows an Arrhenius-type equation:

\(K = A e^{\frac{-E_a}{RT}}\)

Here, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature.

Given the equation:

\(K = \left(2.9 \times 10^{14} \, \text{s}^{-1}\right) e^{\frac{-21000}{T}}\)

By comparing this with the Arrhenius equation:

\(K = A e^{\frac{-E_a}{RT}}\)

We see that \( \frac{E_a}{R} = 21000 \).

To find the activation energy \( E_a \), we use the value of the gas constant \( R = 8.314 \, \text{J/mol K} \):

\(E_a = 21000 \times 8.314 \, \text{J/mol}\)

\(E_a = 174594 \, \text{J/mol} = 174.594 \, \text{kJ/mol}\)

Thus, the correct answer is 3. 174.594 kJ/mol