A reaction has a rate constant of 0.000122/s at 27 °C and 0.228/s at 77 °C. b. What is the value of the rate constant at 17 °C?

Verified step by step guidance

1. Identify that the problem involves the Arrhenius equation, which relates the rate constant \( k \) to temperature \( T \): \( k = A e^{-\frac{E_a}{RT}} \), where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, and \( R \) is the gas constant.
2. Use the Arrhenius equation in its logarithmic form to set up two equations for the two given temperatures: \( \ln k_1 = \ln A - \frac{E_a}{R} \cdot \frac{1}{T_1} \) and \( \ln k_2 = \ln A - \frac{E_a}{R} \cdot \frac{1}{T_2} \).
3. Subtract the first equation from the second to eliminate \( \ln A \) and solve for \( E_a \): \( \ln \frac{k_2}{k_1} = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \).
4. Calculate \( E_a \) using the given rate constants \( k_1 = 0.000122/s \) at \( T_1 = 27 \degree C = 300.15 \text{ K} \) and \( k_2 = 0.228/s \) at \( T_2 = 77 \degree C = 350.15 \text{ K} \).
5. Use the calculated \( E_a \) and the Arrhenius equation to find the rate constant \( k \) at \( 17 \degree C = 290.15 \text{ K} \) by rearranging the equation: \( \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \).

Key Concepts

Rate Constant

The rate constant (k) is a proportionality factor in the rate law of a chemical reaction, indicating the speed of the reaction at a given temperature. It is specific to each reaction and varies with temperature, reflecting how the reaction's rate changes under different conditions.

Arrhenius Equation

The Arrhenius equation describes the temperature dependence of the rate constant, expressed as k = A * e^(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. This equation allows for the calculation of rate constants at different temperatures by relating them through their activation energy.

Temperature Conversion

In chemical kinetics, temperatures must often be converted to Kelvin for calculations. The conversion from Celsius to Kelvin is done by adding 273.15 to the Celsius temperature. Accurate temperature conversion is crucial for applying the Arrhenius equation and determining the rate constant at different temperatures.