The rate constant (k) for a reaction was measured as a function of temperature. A plot of ln k versus 1/T (in K) is linear and has a slope of -7012 K. Calculate the activation energy for the reaction.

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Identify the relationship between the slope of the plot and the activation energy using the Arrhenius equation: \( \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \).

Recognize that the slope of the plot \( \ln k \) versus \( \frac{1}{T} \) is equal to \( -\frac{E_a}{R} \), where \( E_a \) is the activation energy and \( R \) is the universal gas constant.

Use the given slope of the plot, which is -7012 K, and set it equal to \( -\frac{E_a}{R} \).

Solve for the activation energy \( E_a \) by rearranging the equation: \( E_a = -\text{slope} \times R \).

Substitute the value of the universal gas constant \( R = 8.314 \text{ J/mol K} \) into the equation to find \( E_a \).

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주요 개념

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Arrhenius Equation

The Arrhenius equation describes how the rate constant (k) of a chemical reaction depends on temperature (T) and activation energy (Ea). It is expressed as k = A * e^(-Ea/RT), where A is the pre-exponential factor, 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 activation energy barrier.

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. The activation energy can be determined from the slope of a plot of ln k versus 1/T, as it is directly related to the slope in the Arrhenius equation.

Linear Regression in Kinetics

In chemical kinetics, linear regression is often used to analyze the relationship between the natural logarithm of the rate constant (ln k) and the inverse of temperature (1/T). A linear plot indicates a consistent relationship, allowing for the determination of activation energy from the slope. The slope of the line in this context is equal to -Ea/R, where R is the gas constant, enabling the calculation of activation energy when the slope is known.