Problem 109 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant of a reaction at temperature 200 is 10 times less than the rate constant at \(400 \mathrm{~K}\). What is the activation energy (E) of the reaction? \((\mathrm{R}=\) gas constant \()\) (a) \(1842.4 \mathrm{R}\) (b) \(921.2 \mathrm{R}\) (c) \(460.0 \mathrm{R}\) (d) \(230.3 \mathrm{R}\)

Step-by-Step Solution

Answer

The activation energy \(E\) is \(921.2R\).

Step 1: Understanding the problem

We need to find the activation energy \(E\) for a reaction given that the rate constant \(k_1\) at 200 K is 10 times less than the rate constant \(k_2\) at 400 K. We have to use the Arrhenius equation to relate these rate constants to the activation energy.

Step 2: Write the Arrhenius equation

The Arrhenius equation is \( k = A e^{-\frac{E}{RT}} \), where \(k\) is the rate constant, \(A\) is the pre-exponential factor, \(E\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature (in Kelvin).

Step 3: Use the given relationship between rate constants

We are given that \(k_2 = 10k_1\). Using the Arrhenius equation for both conditions:\[ k_1 = A e^{-\frac{E}{R imes 200}} \] \[ k_2 = A e^{-\frac{E}{R imes 400}} \] Set \(k_2 = 10k_1\) to get:\[ A e^{-\frac{E}{R imes 400}} = 10A e^{-\frac{E}{R imes 200}} \]

Step 4: Solve the equation for activation energy (E)

Cancel \(A\) from both sides and take natural logarithm:\[ e^{-\frac{E}{R imes 400}} = 10 e^{-\frac{E}{R imes 200}} \]\[ -\frac{E}{R imes 400} = \ln(10) - \frac{E}{R imes 200} \]Rearrange:\[ \frac{E}{R} \left(\frac{1}{200} - \frac{1}{400}\right) = \ln(10) \] Calculate the numerical factors:\[ \frac{1}{200} - \frac{1}{400} = \frac{1}{200} - \frac{1}{400} = \frac{1}{400}\]Rearrange for E:\[ E = R \times 400 \times \ln(10) \]

Step 5: Calculate the activation energy (E)

Using \( \ln(10) \approx 2.302\), we have:\[ E = R \times 400 \times 2.302 = 921.2R \]

Step 6: Find the correct option

From the given options, the closest match to the calculated activation energy is (b) \(921.2R\).

Key Concepts

Arrhenius Equation

The Arrhenius equation is a crucial part of understanding how chemical reaction rates are influenced by temperature. It is a formula that relates the rate constant of a reaction to several factors:

• Rate constant (\( k \)): Reflects the speed of a reaction.
• Pre-exponential factor (\( A \)): Also known as the frequency factor, which accounts for the frequency of collisions with correct orientation.
• Activation energy (\( E \)): The minimum energy required for a reaction to occur.
• Gas constant (\( R \)): A constant value that links energy, temperature, and moles.
• Temperature (\( T \)): Measured in Kelvin and influences the speed of reactions significantly.

The equation is expressed as:\[ k = A e^{-\frac{E}{RT}} \]This simple exponential relationship shows us how the rate constant decreases as activation energy increases, and increases as temperature rises.
By using the natural exponential, the Arrhenius equation elegantly demonstrates the profound impact these factors have on the speed of chemical reactions.

Rate Constant

The rate constant (\( k \)) is an essential factor in chemical kinetics, defining the speed at which a reaction proceeds. It’s a part of the Arrhenius equation, which ties \( k \) closely to temperature and activation energy. However, it's important to note that the rate constant is highly sensitive to temperature changes.
Different factors influence its value:

• Temperature: As temperature increases, the rate constant generally increases. This results in a faster reaction.
• Catalysts: These substances lower the activation energy, resulting in a higher rate constant.
• Nature of reactants: Different substances have different inherent speeds of reaction, affecting \( k \).

The rate constant is not a universal constant; it changes with different conditions. Understanding \( k \) provides insight into how factors like temperature and catalysts influence reaction speeds.

Temperature Dependence of Reactions

Temperature plays a significant role in determining the rate of chemical reactions. Through the lens of the Arrhenius equation, we can observe how rising temperatures generally lead to an increased rate constant, and consequently, a faster reaction. But why does this happen?

• Higher temperatures give reactant molecules more kinetic energy, leading to more frequent and energetic collisions.
• Increased temperature means a higher proportion of molecules have the necessary activation energy to react.
• As \( T \) rises, the rate constant \( k \) increases exponentially, making reactions noticeably faster.

In summary, the temperature dependence of reactions is not linear but rather exponential, as dictated by the Arrhenius equation. This means small changes in temperature can cause significant changes in reaction rates.