Problem 103 - A Complete Resource Book in Chemistry for JEE Main

Question

When the temperature of a reaction increases from \(27^{\circ} \mathrm{C}\) to \(37^{\circ} \mathrm{C}\), the rate increases by \(2.5\) times, the activation energy in the temperature range is (a) \(70.8 \mathrm{~kJ}\) (b) \(7.08 \mathrm{~kJ}\) (c) \(35.8 \mathrm{~kJ}\) (d) \(14.85 \mathrm{~kJ}\)

Step-by-Step Solution

Answer

The activation energy is approximately \(70.8 \mathrm{kJ/mol}\). Option (a) is correct.

Step 1: Identify the Known Values

The rate of the reaction increases by 2.5 times when the temperature changes from \(27^{\circ} \mathrm{C}\) to \(37^{\circ} \mathrm{C}\). We need to express the temperatures in Kelvin: \(T_1 = 27 + 273 = 300 \text{ K}\) and \(T_2 = 37 + 273 = 310 \text{ K}\).

Step 2: Use the Arrhenius Equation

The Arrhenius equation relates the rate constant \(k\), activation energy \(E_a\), and temperature \(T\): \( k = A \exp\left(-\frac{E_a}{RT}\right) \). The ratio of rate constants at two temperatures can be written as \[ \frac{k_2}{k_1} = \exp\left(\frac{-E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\right) \] where \(k_2 = 2.5k_1\).

Step 3: Set Up the Equation

Substitute the given values into the equation: \[ 2.5 = \exp\left(\frac{-E_a}{8.314}\left(\frac{1}{310} - \frac{1}{300}\right)\right) \]. Here, \(R = 8.314 \text{ J/mol K}\) is the universal gas constant.

Step 4: Simplify the Equation

Calculate the difference \( \frac{1}{T_2} - \frac{1}{T_1} = \frac{1}{310} - \frac{1}{300} \approx -0.000107 \text{ K}^{-1} \). Substitute it back into the equation: \[ 2.5 = \exp\left(\frac{E_a \times 0.000107}{8.314}\right) \].

Step 5: Solve for Activation Energy (Ea)

Take the natural logarithm on both sides: \( \ln(2.5) = \frac{E_a \times 0.000107}{8.314} \). Calculate \( \ln(2.5) \approx 0.916 \). Then, solve for \(E_a\): \[ E_a = \frac{0.916 \times 8.314}{0.000107} \approx 70837 \text{ J/mol} \] or \(70.837 \text{ kJ/mol}\).

Step 6: Select the Correct Option

The calculated value of the activation energy is approximately \(70.8 \text{ kJ/mol}\). This corresponds to option (a).

Key Concepts

Understanding the Arrhenius Equation

The Arrhenius Equation is a critical formula in understanding how reactions proceed at different temperatures. It is expressed as:\[ k = A \exp\left(-\frac{E_a}{RT}\right) \]In this equation, the rate constant \(k\) indicates how fast a reaction goes. Each reaction has a unique pre-exponential factor \(A\), a constant dependent on the nature of the reaction. The activation energy \(E_a\) is the minimum energy needed for the reactants to transform into products. Temperature \(T\) is in Kelvin, and \(R\) is the universal gas constant \(8.314 \, \text{J/mol K}\).

• The Arrhenius Equation shows that higher temperatures lead to larger values of \(k\), meaning the reaction is faster.
• The exponential part \(\exp\left(-\frac{E_a}{RT}\right)\) reflects how temperature impacts the energy barrier that must be overcome.

Using this equation, scientists and students can predict how the reaction rate changes with temperature, which is a crucial aspect of chemical kinetics.

The Temperature Effect on Reaction Rate

Temperature has a profound effect on the speed of chemical reactions. Simply put, reactions generally go faster at higher temperatures. This is because molecules move more quickly and collide more frequently, increasing the chances of overcoming the activation energy barrier. When a reaction's temperature is raised, the following events typically occur:

• Collision Frequency: There's an increase in how often molecules collide.
• Energy Distribution: More molecules have enough energy to get past the activation energy.
• Higher Rate Constants: The rate constants in the Arrhenius Equation increase, which signifies a faster reaction.

These changes make temperature a critical factor in reaction kinetics. It's important to remember that a small temperature change can have a significant effect, as seen in the problem where a mere 10°C increase results in a 2.5-fold increase in reaction rate.

Chemical Kinetics: A Closer Look

Chemical kinetics is the branch of chemistry that examines the speed at which reactions occur and the factors influencing these rates. Key elements in chemical kinetics include:

• Reaction Rate: This measures how quickly reactants are converted into products and can be observed directly or through changes in concentration.
• Rate Laws: These mathematical models describe the relationship between reactant concentrations and the rate of the reaction.
• Mechanism of Reaction: Involves the step-by-step sequence of elementary reactions which lead to the overall reaction.

Using kinetics, chemists can design more efficient chemical processes and gain deeper insights into everything from biological reactions to industrial chemical production. Understanding kinetics helps us harness and control chemical reactions for practical uses, making it a fundamental science discipline.