Problem 4 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Desorption of a gas from metal surface follows first-order kinetics. The rate constant of desorption can be given by Arrhenius equation. If the desorption of hydrogen on manganese is found to increase 10 times on increasing the temperature from 600 to \(1000 \mathrm{~K}\), the activation energy of desorption is (a) \(6.0 \mathrm{kcal} / \mathrm{mol}\) (b) \(6.9 \mathrm{kcal} / \mathrm{mol}\) (c) \(3.0 \mathrm{kcal} / \mathrm{mol}\) (d) \(57.4 \mathrm{kcal} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The activation energy of desorption is approximately 6.9 kcal/mol.

Step 1: Understand the Arrhenius Equation

The Arrhenius Equation describes the temperature dependence of reaction rates. The equation is given by \( k = A \exp(-E_a/(RT)) \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Set up the Ratio of the Rate Constants

Since the desorption rate increases by 10 times, we can set up a ratio of the rate constants at two different temperatures: \( \dfrac{k_2}{k_1} = 10 \), where \( k_1 \) is the rate constant at \( T_1 = 600\ K \) and \( k_2 \) is the rate constant at \( T_2 = 1000\ K \).

Step 3: Apply the Arrhenius Equation to the Rate Constants

By applying the Arrhenius equation to both rate constants we get \ \( \dfrac{A \exp(-E_a/(RT_2))}{A \exp(-E_a/(RT_1))} = 10 \) \ Simplifying this equation gives us \ \( \dfrac{\exp(-E_a/(RT_2))}{\exp(-E_a/(RT_1))} = 10 \)\ or \ \( \exp \left(E_a \left(\dfrac{1}{RT_2}-\dfrac{1}{RT_1}\right)\right) = 10 \).

Step 4: Take the Natural Logarithm to Solve for Activation Energy (E_a)

Taking the natural logarithm of both sides provides a linear equation: \ \( E_a \left(\dfrac{1}{RT_2}-\dfrac{1}{RT_1}\right) = \ln(10) \). Now we can solve for \( E_a \) by substituting the values of \( R \) (Gas constant = 1.987 cal/mol*K), \( T_1 \), and \( T_2 \) into the equation.

Step 5: Substitute the Values and Calculate

Using \( R = 1.987 \text{cal/mol*K} \), \( T_1 = 600\ K \), and \( T_2 = 1000\ K \), we can solve for \( E_a \) as \ \( E_a = \dfrac{\ln(10) \times R}{\dfrac{1}{T_2}-\dfrac{1}{T_1}} = \dfrac{\ln(10) \times 1.987}{\dfrac{1}{1000}-\dfrac{1}{600}} \).

Step 6: Perform the Calculation

After calculating the denominator and multiplying by \( \ln(10) \) and \( R \), we get \( E_a \) in units of cal/mol, which can be converted to kcal/mol by dividing by 1000.

Key Concepts

Activation Energy

Activation energy, often denoted as E_a, is a critical concept in chemical kinetics representing the minimum energy required for a chemical reaction to occur. This energy essentially acts as a barrier that reactant molecules must overcome to transform into products.

In the context of the Arrhenius equation, E_a is the energy that impacts the reaction rate: a high activation energy correlates with slower reactions because fewer molecules have sufficient energy to react. Conversely, a lower E_a suggests that many molecules can overcome this barrier, leading to a faster reaction. When the temperature is increased, more molecules gain the necessary energy to reach or surpass the activation energy, enhancing the reaction rate.

To calculate activation energy, the Arrhenius equation is rearranged, implying that a plot of the natural logarithm of the rate constant (k) against the reciprocal of the temperature (1/T) will yield a straight line with a slope of -E_a/R. By knowing the rate constants at two different temperatures, we can determine E_a as illustrated in the exercise above.

First-Order Kinetics

First-order kinetics is a classification in the field of chemical kinetics describing a reaction where the rate is directly proportional to the concentration of a single reactant. The term 'first-order' indicates that the rate of the reaction depends on the first power of the reactant's concentration.

In mathematical terms, for a reaction with reactant A forming product B, the rate of reaction is described by rate = k[A], where k is the rate constant and [A] is the concentration of reactant A. For first-order reactions, the rate constant k is time-dependent; that is, it tells us how quickly a reactant is consumed over time.

The significance of first-order kinetics in an Arrhenius equation context lies in the fact that it simplifies the analysis of temperature effects on reaction rates. Since the exercise mentions desorption of a gas from a metal surface following first-order kinetics, it means that the reaction rate changes in direct proportion to how the temperature impacts the rate constant k.

Temperature Dependence of Reaction Rates

The temperature dependence of reaction rates is a fascinating area of study in chemical kinetics. It reflects how varying temperatures can significantly alter the speed at which chemical reactions occur. According to the Arrhenius equation, even a small temperature increase can result in a noticeable increase in the reaction rate.

The Arrhenius equation, which includes an exponential term, indicates that reaction rates increase exponentially with temperature. This exponential relationship is because temperature contributes to the kinetic energy of the reacting molecules, increasing the frequency and energy of collisions. With more energy, more molecules are likely to have sufficient kinetic energy to overcome the activation energy barrier.

As shown in the exercise's scenario, the desorption rate of hydrogen on manganese increases 10 times when temperature is raised from 600 K to 1000 K. This demonstrates the profound influence temperature has on reaction rates. To generalize, for most chemical reactions, the rate approximately doubles for every 10-degree Celsius increase in temperature, though this can vary significantly depending on the specific reaction and activation energy involved.