Temperature Dependence of Rate Constant — Revision Notes

⚡ 30-Second Revision

Arrhenius Equation: — $k = A e^{-E_a/RT}$
Linear Form: — $\ln k = \ln A - \frac{E_a}{RT}$
Two Temperatures: — $\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$
Activation Energy ($E_a$): — Minimum energy for reaction, independent of $T$.
Pre-exponential Factor ($A$): — Collision frequency and orientation factor.
Units: — $T$ in Kelvin, $E_a$ in J mol$^{-1}$, $R = 8.314 \text{ J mol}^{-1}\text{ K}^{-1}$.
Plot: — $\ln k$ vs $1/T$ is a straight line with slope $-E_a/R$ (negative slope).

2-Minute Revision

The rate constant ($k$) of a reaction is highly dependent on temperature, a relationship quantified by the Arrhenius equation: $k = A e^{-E_a/RT}$. Here, $A$ is the pre-exponential factor (related to collision frequency and orientation), $E_a$ is the activation energy (the minimum energy barrier for reaction), $R$ is the gas constant, and $T$ is the absolute temperature.

As temperature increases, the fraction of molecules with energy greater than $E_a$ increases exponentially, leading to a faster reaction rate. This is why a $10^{\circ}\text{C}$ rise often doubles or triples the rate.

For calculations, the linear form $\ln k = \ln A - \frac{E_a}{RT}$ is used. A plot of $\ln k$ vs $1/T$ yields a straight line with a negative slope equal to $-E_a/R$, from which $E_a$ can be determined.

For problems involving two different temperatures, use $\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$. Remember to convert temperatures to Kelvin and ensure $E_a$ is in Joules to match $R$'s units.

Catalysts accelerate reactions by lowering $E_a$, making more collisions effective.

5-Minute Revision

The temperature dependence of reaction rates is a cornerstone of chemical kinetics, primarily explained by the Arrhenius equation: $k = A e^{-E_a/RT}$. This equation highlights that the rate constant ($k$) increases exponentially with absolute temperature ($T$).

The term $e^{-E_a/RT}$ represents the fraction of molecules possessing energy equal to or greater than the activation energy ($E_a$), which is the minimum energy barrier for a reaction. As $T$ rises, this fraction increases significantly, leading to more effective collisions and a faster reaction.

The pre-exponential factor ($A$) accounts for the frequency of collisions and the probability of correct molecular orientation.

To experimentally determine $E_a$ and $A$, the Arrhenius equation is linearized by taking the natural logarithm: $\ln k = \ln A - \frac{E_a}{RT}$. This equation is analogous to $y = mx + c$, where a plot of $\ln k$ (y-axis) against $1/T$ (x-axis) yields a straight line. The slope of this line is $-E_a/R$, and the y-intercept is $\ln A$. This graphical method is robust for determining kinetic parameters.

For numerical problems, especially in NEET, the two-point form of the Arrhenius equation is frequently used: $\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$.

This allows calculation of $E_a$ if two rate constants ($k_1, k_2$) at two temperatures ($T_1, T_2$) are known, or calculation of a new rate constant if $E_a$ and one $(k, T)$ pair are given. Critical points for solving problems include: always converting Celsius to Kelvin, ensuring $E_a$ is in J mol$^{-1}$ (if $R = 8.

314 \text{ J mol}^{-1}\text{ K}^{-1}$is used), and being proficient with natural logarithms. Catalysts reduce$E_a$, thereby increasing the rate constant without affecting the overall$\Delta H$ of the reaction.

The empirical rule that reaction rates double or triple for every $10^{\circ}\text{C}$ rise is a direct consequence of this exponential temperature dependence.

Prelims Revision Notes

Temperature Dependence of Rate Constant (Arrhenius Equation)

1. Arrhenius Equation:

* Describes the quantitative relationship between the rate constant ($k$) and absolute temperature ($T$). * Formula: $k = A e^{-E_a/RT}$ * $k$: Rate constant * $A$: Pre-exponential factor (frequency factor), related to collision frequency and orientation. * $E_a$: Activation energy (minimum energy for reaction), always positive. * $R$: Gas constant ($8.314 \text{ J mol}^{-1}\text{ K}^{-1}$) * $T$: Absolute temperature (in Kelvin)

2. Key Concepts:

* **Activation Energy ($E_a$):** Energy barrier that reactants must overcome. Lower $E_a \Rightarrow$ faster reaction. Catalysts lower $E_a$. * **Pre-exponential Factor ($A$):** Reflects the frequency of effective collisions.

* Effect of Temperature: Increasing $T$ increases the kinetic energy of molecules, leading to a larger fraction of molecules having energy $\ge E_a$. This results in an exponential increase in $k$.

* **Temperature Coefficient ($\mu$):** Ratio of rate constants for a $10^{\circ}\text{C}$ temperature difference ($k_{T+10}/k_T$). Typically 2-3.

3. Linear Form for Graphical Analysis:

* Taking natural logarithm: $\ln k = \ln A - \frac{E_a}{RT}$ * This is a straight line ($y = mx + c$) where: * $y = \ln k$ * $x = 1/T$ * Slope ($m$) $= -E_a/R$ (always negative) * Y-intercept ($c$) $= \ln A$ * From slope: $E_a = -R \times \text{slope}$

4. Arrhenius Equation for Two Temperatures:

* Used to calculate $E_a$ or a new $k$ without knowing $A$. * Formula: $\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$ * Alternatively: $\log \left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303R} \left(\frac{T_2 - T_1}{T_1 T_2}\right)$

5. Important Considerations for NEET:

* Units: Always convert $T$ to Kelvin. Ensure $E_a$ is in J mol$^{-1}$ if $R$ is in J mol$^{-1}\text{ K}^{-1}$. * Calculations: Be proficient with natural logarithms ($\ln$) and exponentials ($e^x$). * Catalysts: Lower $E_a$, increasing rate, but do not change $\Delta H$ or equilibrium position. * Conceptual Traps: $E_a$ does NOT change with temperature; the *fraction* of molecules overcoming $E_a$ changes.

Vyyuha Quick Recall

All Reactions Require Temperature, Energy And Kinetics.

Arrhenius Relation: $k = A e^{-E_a/RT}$

A — = Pre-exponential factor
R — = Gas constant
T — = Absolute Temperature
E — = Activation Energy
K — = Rate constant