Among the following factors, the specific reaction rate of a first-order reaction depends on:

Temperature

The correct answer is option 1. Temperature.

Let us look into how each of these factors affects the specific reaction rate (\(k\)) for a first-order reaction.

The specific reaction rate, also known as the rate constant (\(k\)), is a crucial parameter in the rate law of a chemical reaction. For a first-order reaction, the rate law can be expressed as:

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

where:

\(\text{Rate}\) is the rate of the reaction.

\(k\) is the rate constant.

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

The rate constant \(k\) is specific to a particular reaction at a given temperature and does not depend on the concentration of the reactant.

Factors Affecting the Rate Constant (\(k\))

1. Temperature:

The rate constant \(k\) is highly dependent on temperature. According to the Arrhenius equation:

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

where:

\(A\) is the pre-exponential factor (frequency of collisions).

\(E_a\) is the activation energy of the reaction.

\(R\) is the gas constant.

\(T\) is the temperature in Kelvin.

As temperature increases, the exponential term \(e^{-\frac{E_a}{RT}}\) increases, leading to an increase in the rate constant \(k\). This is because higher temperatures provide more energy to the reactant molecules, increasing the fraction of molecules that have enough energy to overcome the activation energy barrier, thus accelerating the reaction.

2. Concentration of Reactant:

For a first-order reaction, the rate of the reaction depends linearly on the concentration of the reactant. However, the rate constant \(k\) itself is independent of the concentration. Changing the concentration of the reactant will change the reaction rate, but it does not affect the value of the rate constant \(k\).

3. Pressure:

For reactions involving gases, pressure can affect the concentration of the gaseous reactants. According to the ideal gas law (\(PV = nRT\)), increasing pressure at a constant temperature increases the concentration of gas. While pressure changes can affect the rate of the reaction by altering the concentration of gaseous reactants, the intrinsic rate constant \(k\) remains unaffected by pressure directly. It is still governed by temperature and the inherent properties of the reaction.

4. Volume:

The volume of the reaction mixture can influence the concentration of reactants if the number of moles remains constant. However, for a first-order reaction, the rate constant \(k\) is not dependent on the volume. Changing the volume can change the concentration of the reactants, which in turn changes the reaction rate, but the rate constant \(k\) itself is not directly influenced by volume.

Summary

Temperature: The specific reaction rate (\(k\)) for a first-order reaction is primarily dependent on temperature. According to the Arrhenius equation, increasing the temperature increases the rate constant \(k\).

Concentration of Reactant: The rate constant \(k\) is independent of the concentration of the reactant in a first-order reaction.

Pressure and Volume: While these factors can influence the concentration of reactants in reactions involving gases, they do not directly affect the rate constant \(k\).

Conclusion

Among the given factors, the specific reaction rate (\(k\)) of a first-order reaction depends on: Temperature

This is because the rate constant \(k\) is a function of temperature, as described by the Arrhenius equation. The concentration of reactants, pressure, and volume influence the overall reaction rate but do not change the intrinsic rate constant \(k\) directly.