The first-order rate constant for the reaction of a particular organic compound with water varies with temperature as follows: Temperature (K) Rate Constant (s⁻¹) 300 3.2 × 10⁻¹¹, 320 1.0 × 10⁻⁹, 340 3.0 × 10⁻⁸, 355 2.4 × 10⁻⁷. From these data, calculate the activation energy in units of kJ/mol.

Verified step by step guidance

Step 1: Use the Arrhenius equation, which relates the rate constant (k) to the temperature (T) and activation energy (Ea): k = A * exp(-Ea/(R*T)), where A is the pre-exponential factor and R is the gas constant (8.314 J/mol·K).

Step 2: Take the natural logarithm of both sides of the Arrhenius equation to linearize it: ln(k) = ln(A) - Ea/(R*T). This equation is in the form of y = mx + c, where y = ln(k), m = -Ea/R, and x = 1/T.

Step 3: Plot ln(k) versus 1/T using the given data points. The slope of the resulting line will be equal to -Ea/R.

Step 4: Calculate the slope of the line from the plot. Use the formula for the slope of a line: slope = (ln(k2) - ln(k1)) / (1/T2 - 1/T1) for any two data points.

Step 5: Solve for the activation energy (Ea) using the slope from Step 4: Ea = -slope * R. Convert the result from J/mol to kJ/mol by dividing by 1000.

Key Concepts

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

Arrhenius Equation

The Arrhenius equation describes how the rate constant of a reaction depends on 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 relationship indicates that as temperature increases, the rate constant typically increases, reflecting a higher likelihood of overcoming the energy barrier for the reaction.

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. Activation energy can be calculated using the slope of a plot of ln(k) versus 1/T, which is derived from the Arrhenius equation.

Temperature Dependence of Reaction Rates

The rate of chemical reactions is significantly influenced by temperature. Generally, as temperature increases, the kinetic energy of molecules also increases, leading to more frequent and effective collisions. This results in an increased reaction rate, which can be quantitatively analyzed using the Arrhenius equation. Understanding this relationship is crucial for calculating activation energy from experimental data, as seen in the provided temperature and rate constant values.

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Related Practice

Textbook Question

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at 520 nm. The reaction occurs in a 1.00-cm sample cell, and the only colored species in the reaction has an extinction coefficient of 5.60 × 10³ M^-1 cm^-1 at 520 nm.

(d) How long does it take for the absorbance to fall to 0.100?

Textbook Question

At 28 C, raw milk sours in 4.0 h but takes 48 h to sour in a refrigerator at 5 C. Estimate the activation energy in kJ>mol for the reaction that leads to the souring of milk.

Textbook Question

The following mechanism has been proposed for the reaction of NO with H₂ to form N₂O and H₂O:

NO(g) + NO(g) → N₂O₂(g)

N₂O₂(g) + H₂(g) → N₂O(g) + H₂O(g)

(a) Show that the elementary reactions of the proposed mechanism add to provide a balanced equation for the reaction.