Problem 22 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

At \(380^{\circ} \mathrm{C}\), the half-life period for the first order decomposition of \(\mathrm{H}_{2} \mathrm{O}_{2}\) is \(360 \mathrm{~min} .\) The energy of activation of the reaction is \(200 \mathrm{~kJ}\) \(\mathrm{mol}^{-1} .\) Calculate the time required for \(75 \%\) decomposition at \(450^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The time for 75% decomposition at 450°C is calculated using rate constants from Arrhenius relation and first-order kinetics.

Step 1: Understand the Problem

We are given the half-life of a first-order reaction at one temperature and need to determine the time required for 75% decomposition at a higher temperature. The reaction's activation energy is also provided.

Step 2: Use Arrhenius Equation

First, we use the Arrhenius equation to find the rate constant (\(k\)) at the higher temperature (450°C): \[k = A e^{-\frac{E_a}{RT}}\]where \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Step 3: Calculate Rate Constant at 380°C

The half-life \(t_{1/2}\) for a first-order reaction is given by \(t_{1/2} = \frac{0.693}{k}\). Calculate \(k\) for 380°C using the given half-life.

Step 4: Convert Temperatures to Kelvin

Convert 380°C and 450°C to Kelvin: 380°C = 653K 450°C = 723K.

Step 5: Calculate New Rate Constant using Arrhenius Equation

Using the Arrhenius equation and the two temperatures, find the new rate constant at 450°C. Use the relation:\[\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)\]Substitute \(k_1\) of 380°C and,\(E_a = 200\, \text{kJ/mol} = 200000 \text{ J/mol}, R = 8.314 \text{ J/mol K}\).

Step 6: Calculate Time for 75% Decomposition

For a first-order reaction, the time \( t\) for any fraction \( f\) decomposed is given by:\[t = \frac{2.303}{k_{new}} \log \left(\frac{1}{1-f}\right)\]For 75% decomposition, \(f = 0.75\). Calculate \(t\) using the rate constant \(k_{new}\) obtained in Step 5.

Key Concepts

Arrhenius equation

The Arrhenius equation is a fundamental formula in chemical kinetics that aids in understanding how reaction rates vary with temperature. It expresses the temperature dependency of the rate constant, denoted as \(k\), which plays a crucial role in predicting reaction behavior. In its equation form,:

• \(k = A e^{-\frac{E_a}{RT}}\)

Here:

• \(k\) is the rate constant
• \(A\) represents the pre-exponential factor, which is related to the frequency of collisions and the probability of favorable orientations
• \(E_a\) stands for the activation energy, or the minimum energy needed for the reaction to occur
• \(R\) is the gas constant, usually \(8.314 \text{ J/mol K}\)
• \(T\) is the temperature in Kelvin

The equation helps quantify the effect of temperature changes on the speed of chemical reactions. It is particularly useful when analyzing reactions at different temperatures, as one can determine the new rate constant \(k\) by comparing with the known rate constant at another temperature.

activation energy

Activation energy \(E_a\) is a critical concept in understanding how chemical reactions proceed. It represents the minimum amount of energy required for reactants to transform into products.
In an energy profile diagram, activation energy is illustrated as the peak between reactants and products:

• This peak symbolizes the energy barrier that must be overcome for a reaction to occur.
• Lower activation energy means reactions can occur with less input of energy, making them faster.

The activation energy is crucial for using the Arrhenius equation, as it directly influences the calculation of the rate constant. Higher activation energies entail slower reactions (given constant temperature) because a greater portion of molecular collisions lack sufficient energy to overcome the activation barrier.
Understanding \(E_a\) allows chemists to predict how changes in conditions like temperature will impact the speed and yield of a reaction. In the exercise, this value is an essential input for estimating reaction behavior at different temperatures.

half-life

The half-life \(t_{1/2}\) is a term often used in the context of first-order reactions, describing the time required for half of a reactant to be consumed. For first-order reactions, it is defined by the formula:

• \(t_{1/2} = \frac{0.693}{k}\)

This relationship means that the half-life is inversely proportional to the rate constant \(k\), indicating that faster reactions have shorter half-lives.
In the context of chemical kinetics, the half-life provides a convenient measure of how quickly a reaction proceeds. It does not depend on the initial concentration of reactants, making it a consistent indicator across different scenarios. This simplicity is why the half-life is frequently used in both academic and practical applications.
In the given exercise, the half-life at a specific temperature (380°C) serves as a stepping stone to find kinetic parameters at another temperature (450°C), highlighting its utility in comparative reaction analysis.

temperature conversion

Temperature conversion is a necessary step in many chemical calculations, especially when using the Arrhenius equation. Temperature must be expressed in Kelvin for most chemical equations because the Kelvin scale starts at absolute zero, providing a consistent foundation for thermal energy measurements.
To convert from Celsius to Kelvin, use the formula:

• \(T(K) = T(°C) + 273.15\)

For the exercise, temperatures are converted as follows:

• 380°C becomes 653K
• 450°C becomes 723K

These conversions ensure all values are suitable for insertion into the Arrhenius equation. Accurate temperature conversion is vital to prevent errors in kinetic calculations, as even small discrepancies alter reaction rate predictions significantly.