A reaction has a rate constant of 0.0117/s at 400.0 K and 0.689/s at 450.0 K. a. Determine the activation barrier for the reaction.

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Identify the given data: rate constants k1 = 0.0117/s at T1 = 400.0 K and k2 = 0.689/s at T2 = 450.0 K.

Use the Arrhenius equation: k = A * e^(-Ea/(RT)), where Ea is the activation energy, R is the gas constant (8.314 J/(mol*K)), and A is the pre-exponential factor.

Take the natural logarithm of both sides of the Arrhenius equation to get: ln(k) = ln(A) - Ea/(RT).

Set up two equations using the given rate constants and temperatures: ln(k1) = ln(A) - Ea/(R*T1) and ln(k2) = ln(A) - Ea/(R*T2).

Subtract the first equation from the second to eliminate ln(A) and solve for Ea: ln(k2/k1) = -Ea/R * (1/T2 - 1/T1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to temperature and activation energy. It is expressed as k = A * e^(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant, and T is the temperature in Kelvin. This equation shows that as temperature increases, the rate constant typically increases, indicating a higher reaction rate.

Activation Energy (Ea)

Activation energy is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to transform into products. A higher activation energy means that fewer molecules have sufficient energy to react at a given temperature, resulting in a slower reaction rate. Understanding Ea is crucial for predicting how temperature changes affect reaction rates.

Temperature Dependence of Reaction Rates

The rate of a chemical reaction is often temperature-dependent, with higher temperatures generally leading to increased reaction rates. This is due to the increased kinetic energy of molecules, which results in more frequent and effective collisions. The relationship between temperature and reaction rate can be quantitatively analyzed using the Arrhenius equation, allowing for the determination of activation energy from rate constants at different temperatures.

Related Practice

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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?

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Textbook Question

Consider these two gas-phase reactions: a. AA(g) + BB(g) → 2 AB(g) b. AB(g) + CD(g) → AC(g) + BD(g) If the reactions have identical activation barriers and are carried out under the same conditions, which one would you expect to have the faster rate?

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Textbook Question

The tabulated data show the rate constant of a reaction measured at several different temperatures. Use an Arrhenius plot to determine the activation barrier and frequency factor for the reaction.

Temperature (K) Rate Constant (1 , s)

300 0.0134

310 0.0407

320 0.114

330 0.303

340 0.757

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