Problem 63 - Chemistry A Molecular Approach

Question

The rate constant \((k)\) for a reaction was measured as a function of temperature. A plot of In \(k\) versus \(1 / T\) (in \(\mathrm{K}\) ) is linear and has a slope of -7445 K. Calculate the activation energy for the reaction.

Step-by-Step Solution

Answer

The activation energy of the reaction is 61892.07 J/mol.

Step 1: Understand the Arrhenius Equation

The Arrhenius equation relates the rate constant (k) of a reaction to the temperature (T) and activation energy (Ea). It can be expressed in a linear form as: ln(k) = ln(A) - (Ea/R)(1/T), where A is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant (8.314 J/mol K), and T is the temperature in Kelvin. The slope of the plot of ln(k) versus 1/T is equal to -Ea/R.

Step 2: Relate the Slope to Activation Energy

From the given information, the slope of the plot is -7445 K. According to the linear form of the Arrhenius equation, this slope is equal to -Ea/R. Therefore, -Ea/R = -7445 K.

Step 3: Calculate the Activation Energy

To find the activation energy (Ea), rearrange the slope equation to solve for Ea: Ea = slope * R. Substitute the given slope and the value of the universal gas constant R into the equation to find Ea: Ea = -7445 K * 8.314 J/mol K.

Step 4: Perform the Multiplication to Get Ea

By multiplying the slope and the gas constant, we get: Ea = -7445 K * 8.314 J/mol K = -61892.07 J/mol. The activation energy of the reaction will be in joules per mole.

Key Concepts

Activation Energy

The term activation energy, represented by the symbol Ea, refers to the minimum amount of energy that reacting species must possess in order to undergo a chemical reaction. Consider it a barrier that reactants need to overcome to transform into products. This concept is crucial as it helps explain why certain reactions occur at different rates under the same conditions.

In the context of our exercise, where the plot of ln(k) versus 1/T yields a straight line with a negative slope, the magnitude of this slope (multiplied by the negative gas constant, R) gives us the activation energy. As we have a slope of -7445 K from the graph, by following the linear form of the Arrhenius equation, we can determine the activation energy. It's the energy barrier that the reactants must overcome to form the products at the given temperature.

Rate Constant

The rate constant, symbolized k, is a proportionality constant that connects the rate of a chemical reaction to the concentration of the reactants. Its value is determined experimentally and can vary with temperature. In essence, k provides the speed at which a reaction proceeds.

According to the Arrhenius equation, the rate constant changes with temperature in an exponential manner, dictated by the activation energy. A higher activation energy generally means a larger temperature effect on the rate constant, resulting in a slower reaction at lower temperatures. Consequently, the Arrhenius equation allows chemists to predict how changes in temperature will affect the reaction rate.

Temperature Dependence of Reaction Rate

Temperature has a significant influence on the rate at which chemical reactions proceed, and this dependency is described by the Arrhenius equation. When temperature increases, the number of reacting molecules with energy equal to or greater than the activation energy rises, facilitating more successful collisions and thus, increasing the rate of reaction.

The exercise we're discussing provides empirical evidence for this temperature dependence, showing that as the temperature changes, the rate constant k also shifts, following a predictable pattern based on the activation energy. The relationship can be visualized with a plot of ln(k) versus 1/T, and the steepness of the line indicates the extent to which temperature impacts the reaction rate. This relationship empowers researchers and industry professionals to tailor reaction conditions for maximum efficiency.