In a reaction, if the rate constant doubles when the temperature increases by 10

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Question: In a reaction, if the rate constant doubles when the temperature increases by 10°C, what is the activation energy (Ea) approximately?

Options:

Correct Answer: 40 kJ/mol

Solution:

Using the Arrhenius equation, Ea can be estimated to be around 40 kJ/mol.

In a reaction, if the rate constant doubles when the temperature increases by 10°C, what is the activation energy (Ea) approximately?

In a reaction, if the rate constant doubles when the temperature increases by 10°C, what is the activation energy (Ea) approximately?

Step 1: Understand the Arrhenius equation, which relates the rate constant (k) to temperature (T) and activation energy (Ea). The equation is k = A * e^(-Ea/(RT)), where A is the pre-exponential factor, R is the gas constant, and T is the temperature in Kelvin.
Step 2: Recognize that if the rate constant doubles (k2 = 2 * k1) when the temperature increases by 10°C, we can set up the equation: 2 * k1 = A * e^(-Ea/(R(T1 + 10))) and k1 = A * e^(-Ea/(RT1)).
Step 3: Take the natural logarithm of both sides of the equation to simplify it: ln(2) = (-Ea/R) * (1/(T1 + 10) - 1/T1).
Step 4: Rearrange the equation to solve for Ea: Ea = R * ln(2) / (1/(T1 + 10) - 1/T1).
Step 5: Use the value of the gas constant R = 8.314 J/(mol·K) and approximate the temperature (T1) in Kelvin. For simplicity, assume T1 is around 298 K (25°C).
Step 6: Calculate the change in the reciprocal of the temperatures: 1/(T1 + 10) - 1/T1 = 1/(308) - 1/(298).
Step 7: Plug the values into the equation to find Ea. After calculation, you will find that Ea is approximately 40 kJ/mol.

Arrhenius Equation – The Arrhenius equation relates the rate constant of a reaction to the temperature and activation energy, showing how temperature affects reaction rates.
Activation Energy (Ea) – The minimum energy required for a chemical reaction to occur, which can be estimated using the change in rate constant with temperature.
Temperature Dependence of Reaction Rates – Understanding how reaction rates increase with temperature and how this is quantitatively described by the Arrhenius equation.