Difficulty: Intermediate | Prerequisites: Parts 1 and 2 of these notes (Kinetics Fundamentals, Rate Laws). Comfort with natural logarithms, linear regression, and the equation of a straight line (y = mx + b).
Parts 1 and 2 showed how to measure rates and extract a rate law at a single temperature. This final piece asks: how does the rate constant k change with temperature, and what does that tell us about the energy barrier the reaction must overcome? The Arrhenius equation connects k to temperature through two parameters, the activation energy (Eₐ) and the frequency factor (A), and an Arrhenius plot is the standard graphical method for extracting both from experimental data.
Plotting ln(k) against 1/T for the iodine clock at seven temperatures gives a straight line. The slope equals -Eₐ/R (giving the activation energy) and the y-intercept equals ln(A) (giving the frequency factor). In this experiment, Eₐ ≈ 48.0 kJ/mol and A ≈ 1.61 × 10³ M⁻¹ s⁻¹, with an R² of 0.9625 confirming a reasonable linear fit.
Arrhenius equation
k = A e^(−Eₐ/RT), where k is the rate constant, A is the frequency factor, Eₐ is the activation energy, R is the gas constant (8.314 J mol⁻¹ K⁻¹), and T is the absolute temperature in kelvin. In simple terms, it says that higher temperatures give molecules more energy to clear the reaction's energy barrier, so k (and therefore the rate) increases.
Activation energy (Eₐ)
The minimum energy that colliding molecules must have for the reaction to proceed. Measured in joules per mole (J/mol) or kilojoules per mole (kJ/mol). Think of it as the height of the energy hill the reactants must climb.
Frequency factor (A)
Also called the pre-exponential factor. It reflects how often molecules collide with the correct orientation. Has the same units as k.
Arrhenius plot
A graph of ln(k) on the y-axis against 1/T on the x-axis. The linearised Arrhenius equation gives a straight line with slope = -Eₐ/R and y-intercept = ln(A).
R² (coefficient of determination)
A measure of how well the linear fit describes the data. R² = 1 is a perfect fit. The value of 0.9625 in this experiment indicates a reasonably good (though not perfect) linear relationship.
The iodine clock was run at seven temperatures using Solution 1 concentrations ([I⁻] = 0.050 M, [S₂O₈²⁻] = 0.050 M). At each temperature, six aliquots were timed and averaged.
T (°C) | T (K) | Avg. time (s) | Rate (M/s) | k (M⁻¹ s⁻¹) | 1/T (K⁻¹) | ln(k) |
|---|---|---|---|---|---|---|
15 | 288 | 173 | 7.25 × 10⁻⁶ | 2.95 × 10⁻⁶ | 0.00347 | -12.73 |
20 | 293 | 134 | 9.33 × 10⁻⁶ | 3.80 × 10⁻⁶ | 0.00341 | -12.48 |
25 | 298 | 92 | 1.36 × 10⁻⁵ | 5.54 × 10⁻⁶ | 0.00336 | -12.10 |
30 | 303 | 60 | 2.07 × 10⁻⁵ | 8.42 × 10⁻⁶ | 0.00330 | -11.68 |
35 | 308 | 43 | 2.92 × 10⁻⁵ | 1.19 × 10⁻⁵ | 0.00325 | -11.34 |
40 | 313 | 47 | 2.68 × 10⁻⁵ | 1.09 × 10⁻⁵ | 0.00319 | -11.43 |
45 | 318 | 24 | 5.21 × 10⁻⁵ | 2.12 × 10⁻⁵ | 0.00314 | -10.77 |
Notice that the 40 °C data point has a longer average time than the 35 °C point. This anomaly (possibly due to a water bath temperature fluctuation) is one reason R² is not closer to 1.
Plot ln(k) on the y-axis and 1/T on the x-axis. A linear regression through these seven points gives:
Slope = -5760.8
y-intercept = 7.2339
R² = 0.9625
The negative slope confirms that k increases with temperature (as 1/T decreases, ln(k) increases).
The slope of the Arrhenius plot equals -Eₐ/R:
-5760.8 = -Eₐ / 8.314 J mol⁻¹ K⁻¹
Eₐ = 5760.8 × 8.314 = 4.79 × 10⁴ J/mol ≈ 48.0 kJ/mol
This is a moderate activation energy, consistent with many ionic reactions in aqueous solution.
Use the Arrhenius equation with a known k and T. At 45 °C (318 K), k = 2.12 × 10⁻⁵ M⁻¹ s⁻¹:
2.12 × 10⁻⁵ = A × e^(-4.80 × 10⁴ / (8.314 × 318))
Solving gives A ≈ 1.61 × 10³ M⁻¹ s⁻¹.
Alternatively, from the y-intercept: ln(A) = 7.2339, so A = e^7.2339 ≈ 1.39 × 10³ M⁻¹ s⁻¹. The slight difference arises because the regression line does not pass exactly through each data point.
Arrhenius equation
k = A\, e^{-E_a / RT}Linearised (log) form
\ln k = -\frac{E_a}{R}\cdot\frac{1}{T} + \ln AThis is in y = mx + b form, where y = ln(k), x = 1/T, m = -Eₐ/R, and b = ln(A).
Activation energy from slope
E_a = -\text{slope} \times RFrequency factor from intercept
A = e^{\text{intercept}}Two-point form (no graph needed)
\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)Useful when you have k at only two temperatures and need to find Eₐ without a full plot.
Students often use °C instead of kelvin in the Arrhenius equation. Temperature must always be in kelvin (K = °C + 273.15).
A common mistake is taking the slope as positive when calculating Eₐ. The slope of an Arrhenius plot is negative (ln(k) decreases as 1/T increases). Eₐ = -slope × R, making Eₐ positive.
Students sometimes think a higher activation energy means a faster reaction. The opposite is true: a higher Eₐ means fewer molecules have enough energy to react at a given temperature, so the reaction is slower.
Confusing R = 8.314 J mol⁻¹ K⁻¹ with R = 0.08206 L atm mol⁻¹ K⁻¹. The Arrhenius equation uses the energy-unit version (8.314 J mol⁻¹ K⁻¹).
⚠️ The Arrhenius plot (ln(k) vs 1/T) and extracting Eₐ from its slope is one of the most commonly examined topics in general chemistry kinetics.
⚠️ Be ready to use the two-point form of the Arrhenius equation when given k at two temperatures without a graph.
⚠️ Know that Eₐ is always positive (it is an energy barrier). If your calculation gives a negative Eₐ, check your signs.
⚠️ Understand what the frequency factor A represents conceptually (collision frequency and orientation), even if the numerical value is hard to interpret.
Fill in the blank: In an Arrhenius plot, the y-axis is ______ and the x-axis is ______. (ln(k); 1/T)
True or false: The activation energy can be negative. (False; Eₐ is always positive.)
Fill in the blank: The slope of an Arrhenius plot equals ______. (-Eₐ/R)
True or false: Temperature in the Arrhenius equation must be in °C. (False; it must be in kelvin.)
Fill in the blank: If the slope of an Arrhenius plot is -5760.8, then Eₐ = ______ kJ/mol. (5760.8 × 8.314 / 1000 ≈ 47.9 kJ/mol)
Q: Given a slope of -5760.8 from an Arrhenius plot, calculate the activation energy in kJ/mol.
A: Eₐ = -slope × R = 5760.8 × 8.314 J mol⁻¹ K⁻¹ = 4.79 × 10⁴ J/mol = 47.9 kJ/mol (the lab report rounds to 48.0 kJ/mol).
Q: Using k = 2.12 × 10⁻⁵ M⁻¹ s⁻¹ at 318 K and Eₐ = 48.0 kJ/mol, calculate the frequency factor A.
A: k = A e^(-Eₐ/RT). Rearranging: A = k / e^(-Eₐ/RT) = 2.12 × 10⁻⁵ / e^(-48000 / (8.314 × 318)) = 2.12 × 10⁻⁵ / e^(-18.14) ≈ 2.12 × 10⁻⁵ / 1.32 × 10⁻⁸ ≈ 1.61 × 10³ M⁻¹ s⁻¹.
Q: The 40 °C data point shows a longer average time than the 35 °C point. What might cause this?
A: Probable causes include the water bath temperature being unstable or not reaching exactly 40 °C before mixing, operator timing error, or incomplete mixing. In the Arrhenius plot this anomaly pulls the data point away from the best-fit line and lowers R².
Q: If a reaction has an Eₐ of 100 kJ/mol compared to this reaction's 48 kJ/mol, which is faster at the same temperature (all else being equal)?
A: The reaction with Eₐ = 48 kJ/mol is faster. A lower activation energy means a greater fraction of molecules have enough energy to react, so the rate constant k is larger at any given temperature.
Q: Using the two-point form, estimate Eₐ from the 15 °C and 45 °C data alone (k₁ = 2.95 × 10⁻⁶ at 288 K, k₂ = 2.12 × 10⁻⁵ at 318 K).
A: ln(k₂/k₁) = Eₐ/R × (1/T₁ - 1/T₂). ln(2.12 × 10⁻⁵ / 2.95 × 10⁻⁶) = ln(7.19) = 1.973. (1/288 - 1/318) = 0.000327. Eₐ = 1.973 × 8.314 / 0.000327 ≈ 50,100 J/mol ≈ 50.1 kJ/mol. This is close to the 48.0 kJ/mol from the full seven-point regression.
The Arrhenius equation connects kinetics to thermodynamics: Eₐ is related to the enthalpy of the transition state in transition-state theory. In catalysis, a catalyst works by providing an alternative pathway with a lower Eₐ, which you can verify experimentally by comparing Arrhenius plots with and without the catalyst. In biochemistry, enzymes are biological catalysts that dramatically reduce Eₐ, and enzyme denaturation at high temperatures shows the limits of the Arrhenius model.
Arrhenius equation, Arrhenius plot, activation energy calculation, frequency factor pre-exponential, ln(k) vs 1/T, temperature dependence rate constant, Eₐ from slope, two-point Arrhenius, R² linear regression kinetics, iodine clock temperature, rate constant vs temperature, collision theory, transition state theory, kJ/mol activation energy, water bath kinetics experiment