Question

Two reactions \(\mathrm{X} \rightarrow\) Products and \(\mathrm{Y} \rightarrow\) products have rate constant \(\mathrm{k}_{\mathrm{x}}\) and \(\mathrm{k}_{\mathrm{Y}}\) at temperature \(\mathrm{T}\) and activation energies \(\mathrm{E}_{\mathrm{x}}\) and \(\mathrm{E}_{\mathrm{Y}}\) respectively. If \(\mathrm{k}_{\mathrm{x}}>\) \(\mathrm{k}_{\mathrm{r}}\) and \(\mathrm{E}_{\mathrm{x}}<\mathrm{E}_{\mathrm{Y}}\) and assuming that for both the reaction is same, then a. At lower temperature \(\mathrm{k}_{\mathrm{Y}}>\mathrm{k}_{\mathrm{x}}\) b. At higher temperature \(\mathrm{k}_{\mathrm{x}}\) will be greater than \(\mathrm{k}_{\mathrm{y}}\) c. At lower temperature \(\mathrm{k}_{\mathrm{x}}\) and \(\mathrm{k}_{\mathrm{Y}}\) will be close to each other in magnitude d. At temperature rises, \(\mathrm{k}_{\mathrm{x}}\) and \(\mathrm{k}_{\mathrm{Y}}\) will be close to each other in magnitude

Short answer

Correct choice is b: At higher temperature, \( k_x \) will be greater than \( k_y \).

Step-by-step solution

Understanding the Arrhenius Equation

The Arrhenius equation gives the temperature dependence of reaction rates: \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the frequency factor, \( E_a \) is the activation energy, and \( RT \) is the product of the gas constant \( R \) and temperature \( T \). Lower activation energy \( E_x < E_y \) means that for higher temperatures, reaction X will be more favorable.

Comparing Rate Constants at Low Temperatures

Given the lower activation energy \( E_x < E_y \) and initially \( k_x > k_y \), at lower temperatures, \( k_y \) might surpass \( k_x \) due to the temperature effect on \( k \) being less significant for reaction X. However, since \( k_x \) starts higher, it's unlikely for \( k_y \) to exceed \( k_x \) drastically unless temperature changes significantly.

Evaluating Rate Constants at High Temperatures

At high temperatures, because \( k_x > k_y \) initially and \( E_x < E_y \), \( k_x \) will remain greater due to the stronger influence of the higher initial rate constant and lower activation energy.

Assessing the Rate Constant Proximity

Evaluate which scenario might cause the rate constants to be close. Since \( E_x < E_y \), as temperature increases, \( k_x \) will continue to grow rapidly, staying greater than \( k_y \) and not close in magnitude. At low temperatures, initial difference (\( k_x > k_y \)) and impact due to \( E_x < E_y \) mean they likely won't converge closely.

Key concepts

Arrhenius Equation

In chemical kinetics, the Arrhenius equation plays a crucial role in understanding how temperature affects reaction rates. The equation is given by:\[k = A e^{-\frac{E_a}{RT}}\]Here, \(k\) represents the rate constant, \(A\) is the frequency factor indicating how often molecules collide with the correct orientation, \(E_a\) is the activation energy, \(R\) stands for the gas constant, and \(T\) is the temperature expressed in Kelvin. This equation implies that as the energy barrier \(E_a\) decreases or as the temperature \(T\) increases, the rate constant \(k\) will increase.
The Arrhenius equation helps explain why certain reactions occur faster than others, primarily due to differences in activation energy or the impact of temperature on these processes.

Rate Constant Comparison

Comparing rate constants, \(k_x\) and \(k_y\), from reactions with different activation energies and initial values is insightful for predicting reaction behaviors across temperatures. If initially, \(k_x\) is greater than \(k_y\), it suggests reaction X is faster at the same temperature due to its lower activation energy. Hence, when analyzing how these constants behave, it is essential to consider:

• Initial differences: The starting magnitude of \(k_x\) being larger can set the pace for trajectory at various temperatures.
• Impact of \(E_a\): With \(E_x < E_y\), \(k_x\) tends to remain larger, especially as temperature rises, because the activation barrier is easier to overcome.

By evaluating these factors over temperature changes, you can predict which reaction will proceed more swiftly under given conditions.

Activation Energy Effects

Activation energy \(E_a\) is a pivotal concept that determines the susceptibility of a reaction rate constant \(k\) to change with temperature. Lower activation energy like \(E_x\) means:

• Reactions can happen more readily as less energy is needed to reach the transition state.
• It exhibits a smaller temperature dependence initially but allows for rapid increases in \(k\) as temperature rises.

For our two reactions, reaction X with \(E_x < E_y\) would inherently see a greater change in its rate constant with temperature variation, particularly at higher temperatures, solidifying \(k_x\)'s dominance over \(k_y\). Understanding these effects aids in controlling and optimizing reactions by manipulating conditions to either accelerate or decelerate the process as needed.

Temperature Dependence of Reactions

The way reaction rates change with temperature is fundamental to chemical kinetics and can be largely predicted using the Arrhenius equation. High temperatures generally increase reaction rates because:

• Molecules move faster, resulting in more frequent and energetic collisions.
• The exponential factor in the Arrhenius equation increases drastically, especially for reactions with lower activation energies.

In our case, with lower \(E_x\), reaction X will experience a larger increase in \(k_x\) compared to \(k_y\) as temperatures rise.
This underscores why sometimes simply raising the temperature of a system can efficiently speed up a reaction, a technique frequently used in industrial chemical processes to achieve desired reaction rates. However, careful control is essential to prevent unintended consequences or decompositions when temperatures are too high.