Arrhenius Equation (Graphical) Equation, Derivation & Rearrangements

Core idea

Overview

The Arrhenius equation describes the mathematical relationship between the rate constant of a chemical reaction and its absolute temperature. It illustrates how the frequency factor and activation energy determine the temperature dependence of reaction kinetics.

When to use: Apply this model when investigating how changes in temperature influence the speed of a chemical reaction. It is the standard tool for calculating activation energy from experimental data where the rate constant is measured at several temperatures.

Why it matters: This equation is essential for predicting the stability of chemicals and food products over time. It allows chemical engineers to optimize temperature conditions in industrial reactors to balance yield and energy costs.

Symbols

Variables

A = Pre-exponential Factor, $E_{a}$ = Activation Energy, R = Gas Constant, T = Temperature, k = Rate Constant

A

Pre-exponential Factor

s^{-} 1

E_{a}

Activation Energy

J/mol

R

Gas Constant

J/molK

T

Temperature

K

k

Rate Constant

s^{-} 1

k_{1}

Rate Constant 1

s^{-} 1

T_{1}

Temperature 1

K

k_{2}

Rate Constant 2

s^{-} 1

T_{2}

Temperature 2

K

Walkthrough

Derivation

Formula: Arrhenius Equation

Shows how the rate constant depends on temperature and activation energy.

Activation energy $E_{a}$ is approximately constant over the temperature range considered.
Pre-exponential factor A is approximately constant over the temperature range considered.

1

State the Equation:

k increases with T because the exponential factor becomes less negative; larger $E_{a}$ makes k smaller at a given T.

k = A e^{- \frac{E _{a}}{R T}}

2

Linear (Log) Form:

Plotting $ln$ k against 1/T gives a straight line with gradient - $E_{a}$ /R and intercept $ln$ A.

ln k = ln A - \frac{E _{a}}{R} (\frac{1}{T})

Result

ln k = ln A - \frac{E _{a}}{R} (\frac{1}{T})

Source: OCR A-Level Chemistry A — Reaction Rates

Free formulas

Rearrangements

Solve for $A$

Make A the subject

A = k e^{E_{a} / (R T)}

Rearrange the Arrhenius equation to make the pre-exponential factor, $A$ , the subject. This involves isolating $A$ by dividing both sides by the exponential term and then simplifying the expression using index laws.

Difficulty: 2/5

Solve for $E_{a}$

Make Ea the subject

E_{a} = - R T ln (\frac{k}{A})

To make $E_{a}$ the subject of the Arrhenius equation, first isolate the exponential term by dividing by the pre-exponential factor $A$ . Then, take the natural logarithm of both sides to remove the base $e$ .

Difficulty: 2/5

Solve for $T$

Make T the subject

T = - \frac{E _{a}}{R ln ( \frac{k}{A} )}

Rearrange the Arrhenius Equation to make Temperature (T) the subject. This involves isolating the exponential term, taking the natural logarithm, and then performing algebraic manipulations to solve for T.

Difficulty: 3/5

Solve for $R$

Make R the subject

R = - \frac{E _{a}}{T ln ( \frac{k}{A} )}

Rearrange the Arrhenius equation to make the gas constant, R, the subject. This involves isolating the exponential term, taking the natural logarithm of both sides, and then performing algebraic manipulations to solve for R.

Difficulty: 2/5

Solve for $E_{a}$

Make Ea the subject

E_{a} = \frac{R ln ( k _{2} / k _{1} )}{1/ T _{1} - 1/ T _{2}}

Derive the two-point form of the Arrhenius equation to solve for activation energy ( $E_{a}$ ) by taking natural logarithms at two different temperatures and subtracting the resulting equations.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows an exponential growth curve where the rate constant rises as temperature increases and eventually levels off toward the value of the pre-exponential factor A. For a chemistry student, this shape demonstrates that at low temperatures the reaction is significantly restricted, while at high temperatures the rate constant approaches its theoretical maximum. The most important feature of this curve is that the rate constant never reaches zero, meaning that even at very low temperatures there is a mathematical probability for the reaction to occur.

Graph type: exponential

Why it behaves this way

Intuition

Imagine a reaction as molecules trying to climb over an energy hill (activation energy); only molecules with sufficient kinetic energy (determined by temperature)

k

Rate constant of the reaction

Quantifies the speed of a chemical reaction; a larger 'k' means a faster reaction.

A

Pre-exponential factor (or frequency factor)

Represents the frequency of correctly oriented collisions between reactant molecules, assuming no energy barrier.

E_{a}

Activation energy

The minimum energy barrier that reactant molecules must overcome to form products; higher

E_{a}

means fewer successful collisions.

R

Ideal gas constant

A proportionality constant that relates energy to temperature, allowing

E_{a}

and T to be compared on an energy scale.

T

Absolute temperature

A measure of the average kinetic energy of the reactant molecules; higher T means more energetic collisions.

Signs and relationships

-E_a/(RT): The negative sign in the exponent, combined with the inverse temperature dependence (1/T), reflects the Boltzmann distribution. It indicates that only a fraction of molecules possess energy greater than the activation

Free study cues

Insight

Canonical usage

Units for activation energy ( $E_{a}$ ) and the ideal gas constant (R) must be consistent (e.g., both in Joules per mole) and temperature (T) must be in Kelvin to ensure the exponent is dimensionless.

Common confusion

A common mistake is using temperature in Celsius instead of Kelvin, or mixing units for activation energy (e.g., kJ mol-1) and the gas constant (J mol-1 K-1) without proper conversion.

Dimension note

The exponent $E_{a}$ /(RT) must be dimensionless. This requires $E_{a}$ and RT to have the same units (e.g., J mol-1).

Unit systems

$k$ Varies by reaction order (e.g., s-1 for first order, M-1s-1 for second order) · The unit of the rate constant k depends on the overall order of the reaction. The unit of the pre-exponential factor A is always the same as k.

$A$ Varies by reaction order (e.g., s-1 for first order, M-1s-1 for second order) · The unit of the pre-exponential factor A is always the same as the rate constant k.

$E_{a}$ J mol-1 (or kJ mol-1 requiring conversion) · Activation energy is commonly reported in kJ mol-1, which must be converted to J mol-1 to be consistent with the SI unit of R.

$T$ K · Temperature must always be in Kelvin (absolute temperature).

Ballpark figures

Quantity:

One free problem

Practice Problem

A reaction has an activation energy (Ea) of 80,000 J/mol and a pre-exponential factor A = 2.5×10^13 s^-1. Calculate the rate constant k at 350 K. (R = 8.314 J/mol·K)

Pre-exponential Factor25000000000000 s^-1

Activation Energy80000 J/mol

Gas Constant8.314 J/molK

Temperature350 K

Solve for: $k$

Hint: Use k = A × exp(-Ea / (RT)). Calculate the exponent first.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating how much faster a reaction runs at higher T, Arrhenius Equation (Graphical) is used to calculate Rate Constant from Pre-exponential Factor, Activation Energy, and Gas Constant. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure temperature is converted to Kelvin before calculation.
Check that activation energy (Ea) and the gas constant (R) use the same energy units, typically Joules.
A plot of ln(k) versus 1/T yields a straight line with a slope equal to -Ea/R.
The pre-exponential factor A represents the frequency of collisions with the correct molecular orientation.

Avoid these traps

Common Mistakes

Using Celsius instead of Kelvin.
Mixing kJ and J for Ea.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Shows how the rate constant depends on temperature and activation energy.

Apply this model when investigating how changes in temperature influence the speed of a chemical reaction. It is the standard tool for calculating activation energy from experimental data where the rate constant is measured at several temperatures.

This equation is essential for predicting the stability of chemicals and food products over time. It allows chemical engineers to optimize temperature conditions in industrial reactors to balance yield and energy costs.

Using Celsius instead of Kelvin. Mixing kJ and J for Ea.

When estimating how much faster a reaction runs at higher T, Arrhenius Equation (Graphical) is used to calculate Rate Constant from Pre-exponential Factor, Activation Energy, and Gas Constant. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure temperature is converted to Kelvin before calculation. Check that activation energy (Ea) and the gas constant (R) use the same energy units, typically Joules. A plot of ln(k) versus 1/T yields a straight line with a slope equal to -Ea/R. The pre-exponential factor A represents the frequency of collisions with the correct molecular orientation.

References

Sources

Atkins' Physical Chemistry
Wikipedia: Arrhenius equation
IUPAC Gold Book: Arrhenius equation
NIST CODATA
Atkins' Physical Chemistry, 11th Edition
IUPAC Gold Book
Atkins' Physical Chemistry, 11th Edition, Peter W. Atkins, Julio de Paula, James Keeler
IUPAC Gold Book (Compendium of Chemical Terminology)

Arrhenius Equation (Graphical)

Overview

Variables

Derivation

State the Equation:

Linear (Log) Form:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Rate law

Frequently Asked Questions

Sources