The plot of Ink and 1/T is given in the figure what is its slope?

The correct answer is option 1.  \(-\frac{E_a}{R}\).

The Arrhenius equation describes the temperature dependence of reaction rates. It states that the rate constant (\(k\)) of a reaction is exponentially related to the inverse of the temperature (\(T\)):

\(k = Ae^{-\frac{E_a}{RT}} \)

Where:

\( A \) is the pre-exponential factor, which represents the frequency of collisions between reactant molecules.

\( E_a \) is the activation energy, which represents the minimum energy required for a reaction to occur.

\( R \) is the gas constant.

\( T \) is the absolute temperature in Kelvin.

Taking the natural logarithm of both sides of the equation, we get:

\( \ln(k) = \ln(A) - \frac{E_a}{RT} \)

This equation has the form of a straight line equation, \(y = mx + c\), where:

\( y = \ln(k) \),

\( m = -\frac{E_a}{R} \),

\( x = \frac{1}{T} \), and

\( c = \ln(A) \).

So, when we plot \(\ln(k)\) against \(\frac{1}{T}\), we should get a straight line with slope \(m = -\frac{E_a}{R}\). This slope represents the ratio of activation energy to the gas constant (\(R\)).

Hence, the correct option is: 1. \(-\frac{E_a}{R}\)

This relationship is fundamental in understanding how temperature affects reaction rates and is often used in kinetics studies to determine activation energies.