Question

How does temperature affect k, the rate constant? Explain.

Short answer

The rate constant (k) is affected by temperature through the Arrhenius Equation: \(k = A \times e^{-\frac{Ea}{RT}}\). As temperature increases, the exponential factor (\(e^{-\frac{Ea}{RT}}\)) increases, leading to an increase in the rate constant and a faster reaction. The activation energy (Ea) determines how sensitive the rate constant is to temperature changes, with higher activation energy making the reaction more sensitive to temperature changes.

Step-by-step solution

Understanding the Arrhenius Equation

The connection between temperature and the rate constant (k) is given by the Arrhenius Equation: \[k = A \times e^{-\frac{Ea}{RT}}\] where: - k is the rate constant - A is the pre-exponential factor (a constant) - Ea is the activation energy (energy needed for a reaction to occur) - R is the ideal gas constant (8.314 J/mol K) - T is the temperature in Kelvin

Effect of temperature on k

In the Arrhenius equation, the rate constant k depends on the temperature through the exponential factor: \[e^{-\frac{Ea}{RT}}\] This factor is a dimensionless number between 0 and 1, which decreases as the activation energy (Ea) increases and/or the temperature (T) decreases. Conversely, this factor increases as the temperature (T) increases.

Evaluating the exponential factor

When the temperature is low, the exponential factor becomes closer to 0. This means that the rate constant (k) is also lower, leading to a slower reaction. As the temperature increases, the exponential factor becomes closer to 1, increasing the rate constant (k) and making the reaction faster. This is why temperature plays a significant role in determining the rate of a reaction.

Activation energy and temperature sensitivity

The activation energy (Ea) determines how sensitive a reaction's rate constant (k) is to temperature changes. Reactions with high activation energy are more sensitive to temperature changes, meaning that small changes in temperature result in significant changes in the rate constant. Reactions with low activation energy are less sensitive to temperature changes, meaning that changes in temperature have a smaller effect on the rate constant. In summary, the rate constant (k) is affected by temperature through the Arrhenius Equation. As temperature increases, the exponential factor in the Arrhenius Equation increases, leading to an increase in the rate constant and a faster reaction. The activation energy of the reaction determines how sensitive the rate constant is to temperature changes.