Problem 92 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The rate of a reaction is approximately doubled for every \(10^{\circ} \mathrm{C}\) rise in temperature. If the temperature is raised by \(50^{\circ} \mathrm{C}\), the reaction rate increases by about (a) 10 times (b) 16 times (c) 32 times (d) 64 times

Step-by-Step Solution

Answer

The reaction rate increases by 32 times for a 50°C rise in temperature.

Step 1 - Understand the Given Information

The exercise provides the rule that the rate of a chemical reaction approximately doubles for every increase of 10°C in temperature. This means that with each 10°C increase, we can think of the rate as being multiplied by a factor of 2.

Step 2 - Calculate the Total Increase for 50°C

Since the rate doubles every 10°C, we need to determine how many times this doubling occurs over a 50°C increase. Since 50°C is equal to 5 intervals of 10°C, we multiply the rate by 2 for each of the 5 intervals.

Step 3 - Apply the Doubling Rule

To find the overall factor by which the reaction rate increases over a 50°C temperature increase, raise the number 2 to the power of 5, as we have 5 intervals of doubling. The calculation is as follows: \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\).

Step 4 - Select the Answer

From the calculation performed in Step 3, we determine that the reaction rate increases by a factor of 32 over a 50°C increase. Therefore, the correct answer is (c) 32 times.

Key Concepts

Arrhenius Equation

The Arrhenius Equation plays a crucial role in understanding the temperature dependence of reaction rates. It is expressed as:
\[ k = A e^{-\frac{E_a}{RT}} \]
where \( k \) is the rate constant, \( A \) is the frequency factor reflecting the number of times reactants approach each other per unit time, \( E_a \) is the activation energy necessary for the reaction to occur, \( R \) is the gas constant, and \( T \) is the absolute temperature in kelvins. The equation shows that an increase in temperature (\( T \)) leads to an exponential increase in the rate constant (\( k \)), thus speeding up the reaction. This explains why a seemingly small temperature increase can result in a significant increase in reaction rate. Understanding the implications of this equation helps students grasp why the reaction rate increases by a factor of 32 when the temperature is raised by 50°C, as calculated in the exercise.

Reaction Kinetics

Fundamental to the study of chemical reactions, Reaction Kinetics involves the analysis of rates at which reactions occur and the steps through which they proceed. Key factors affecting reaction rates include the concentration of reactants, surface area, temperature, and catalysts. These factors either provide more opportunities for reactant particles to collide or lower the energy required for the reaction. Students should understand that the rate of a reaction is measured by the change in concentration of reactants or products over time. The temperature coefficient discussed in the exercise exemplifies how kinetics principles can be applied to predict the impact of temperature changes on reaction rates. Reaction kinetics emphasizes that not all collisions result in a reaction; only those with sufficient energy and proper orientation lead to product formation.

Temperature Coefficient of Reaction Rate

The Temperature Coefficient of Reaction Rate is a numerical value that quantifies the change in the rate of a reaction resulting from a temperature change. Typically, for many reactions, the rate approximately doubles for every 10°C increase in temperature, which can also be expressed using the rule of thumb known as the Q10 coefficient. The Q10 coefficient is described as:
\[ Q10 = \left( \frac{\text{Rate at } T+10}{\text{Rate at } T} \right) \]
where \( T \) is the initial temperature. Using the Q10 coefficient can simplify complex calculations involving the Arrhenius Equation to understand the impact of temperature on reaction rates. The solution to the given problem showcases this: raising temperature by 50°C increased the reaction rate by 32 times (as per the calculation \(2^5\)), clearly illustrating the exponential nature of this dependence. Educators can explain this concept by emphasizing that with every 10°C rise, the kinetic energy of molecules increases, leading to more frequent and effective collisions, thereby increasing the reaction rate.