The specific rate constant of a first order reaction depends on the

Temperature

The correct answer is option 4. Temperature.

In a first-order reaction, the rate of the reaction depends linearly on the concentration of one reactant, but the rate constant (\(k\)) is independent of the concentration of reactants or products. The rate law for a first-order reaction is generally written as:

\(\text{Rate} = k [A]\)

Where:

\([A]\) is the concentration of the reactant,

\(k\) is the rate constant (specific rate constant).

The rate constant \(k\) is a measure of how fast the reaction proceeds. For first-order reactions, the units of \(k\) are \( \text{time}^{-1} \), for example, \( s^{-1} \).

Dependence of the rate constant:

The rate constant (\(k\)) for a reaction depends only on temperature and, to some extent, the presence of a catalyst (if applicable). It does not depend on the concentration of reactants, products, or time. The reason for this dependence is related to how temperature influences the kinetic energy of molecules and the activation energy of the reaction.

Arrhenius Equation

The relationship between the rate constant \(k\) and temperature is described by the Arrhenius equation:

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

Where:

\(k\) = rate constant,

\(A\) = frequency factor or pre-exponential factor (related to the frequency of collisions and the orientation of reactant molecules),

\(E_a\) = activation energy (the minimum energy needed for the reaction to occur),

\(R\) = universal gas constant (8.314 J/mol·K),

\(T\) = temperature in Kelvin.

This equation shows that the rate constant \(k\) increases as the temperature \(T\) increases. This is because higher temperatures provide more energy to the reacting molecules, increasing the likelihood that they will overcome the activation energy barrier, leading to faster reaction rates.

Effect of Various Factors on the Rate Constant:

1. Concentration of the Reactant:

In a first-order reaction, the rate depends on the concentration of the reactant, but the rate constant \(k\) itself is not influenced by the concentration.

For example, if you double the concentration of the reactant, the rate of the reaction doubles, but the value of \(k\) remains unchanged.

2. Concentration of the Product:

The concentration of the product does not influence the rate constant \(k\), as the rate law for a first-order reaction only involves the reactants.

3. Time:

Time is also not a factor that affects the rate constant \(k\). The rate constant remains constant over time for a given temperature and reaction conditions. The rate of the reaction changes over time as the concentration of the reactants decreases, but this change is a result of the reaction progressing, not a change in \(k\).

4. Temperature:

The rate constant \(k\) is highly dependent on temperature. As temperature increases, molecules have more kinetic energy, which means more of them will have enough energy to overcome the activation energy barrier, resulting in a faster reaction and a larger rate constant. This dependence on temperature is reflected in the exponential factor \(e^{\frac{-E_a}{RT}}\) in the Arrhenius equation. As \(T\) increases, the value of \(k\) increases exponentially, reflecting the strong influence of temperature on reaction rates.

Summary:

The rate constant for a first-order reaction depends on temperature because increasing temperature allows more reactant molecules to have enough energy to surpass the activation energy.

The rate constant does not depend on the concentration of reactants, products, or time.

The relationship between temperature and the rate constant is captured by the Arrhenius equation, which shows that the rate constant increases as temperature rises.

Thus, the correct answer is: 4. Temperature.