The decomposition of hydrocarbon follows the equation $k = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}$ Calculate $E_a$.

The decomposition of hydrocarbon follows the equation k = (4.5×10¹¹s⁻¹)e-28000K/T Calculate Ea.

Exercises · 3.26

k = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}

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Concept understanding — Arrhenius Equation

The Arrhenius Equation Plot: Why Temperature Changes Reaction Speed

You already know that heating things up makes reactions go faster. A cold chai takes forever to dissolve sugar; hot chai does it in seconds. But how much faster? And is there a pattern that holds for every reaction?

That pattern is the Arrhenius equation, and plotting it in a clever way reveals something fundamental about how molecules need to collide to react.

The core idea: an energy barrier

Imagine a ball sitting in a valley. To get to the next valley, it must first be pushed up over a hill. That hill is the activation energy ( $E_a$ ) — the minimum energy two molecules need to have when they collide, for the reaction to happen.

At a low temperature, most molecules move slowly. Only a tiny fraction have enough energy to climb that hill. Raise the temperature, and suddenly many more molecules have the required energy. The fraction of molecules with energy $\ge E_a$ is given by the Boltzmann distribution:

\text{fraction} = e^{-E_a / RT}

where $R$ is the gas constant and $T$ is the absolute temperature (in Kelvin). This exponential is the heart of the story.

The Arrhenius equation (precise statement)

The rate constant $k$ of a reaction depends on temperature as:

k = A \, e^{-E_a / RT}

$k$ = rate constant (how fast the reaction proceeds)
$A$ = pre-exponential factor (frequency of collisions, times a steric factor — how often molecules hit in the right orientation)
$E_a$ = activation energy (J/mol or kJ/mol)
$R$ = 8.314 J/(mol·K)
$T$ = temperature in Kelvin

Important

$k$ is not the reaction rate itself — it's the proportionality constant in the rate law. But for a fixed concentration, a larger $k$ means a faster reaction.

Why plot it? The linear trick

The equation $k = A e^{-E_a / RT}$ is exponential in $1/T$ . That's hard to eyeball. But take the natural logarithm of both sides:

\ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T}

This is the equation of a straight line:

y = c + m x

where:

$y = \ln k$
$x = 1/T$
slope $m = -E_a / R$
intercept $c = \ln A$

So if you measure $k$ at several temperatures and plot $\ln k$ versus $1/T$ , you get a straight line — provided the reaction follows Arrhenius behaviour (most do, over moderate temperature ranges).

Tip

Always use Kelvin for $T$ . Celsius will give you a curved mess because $1/T$ is not linear in Celsius.

What the plot tells you

From the slope, you get $E_a$ :

E_a = - (\text{slope}) \times R

A steep negative slope means a large $E_a$ — the reaction is very sensitive to temperature. A shallow slope means a small $E_a$ — temperature doesn't affect it much.

From the intercept, you get $A$ :

A = e^{\text{intercept}}

This tells you about the collision frequency and orientation factor. A high $A$ means molecules are colliding often and in the right geometry.

A typical Arrhenius plot looks like this

$T$ (K)	$k$ (s⁻¹)	$1/T$ (K⁻¹)	$\ln k$
300	0.0012	0.00333	-6.72
310	0.0028	0.00323	-5.88
320	0.0061	0.00313	-5.10
330	0.0125	0.00303	-4.38

Plot $\ln k$ (y-axis) vs $1/T$ (x-axis). The points fall on a straight line. Draw the best-fit line, measure its slope, and compute $E_a$ . …

Why this formula?

Arrhenius Equation: Why It Holds

The Arrhenius equation is not a guess — it emerges from a deep physical picture of how molecules react. Let's build that picture step by step.

1. The Core Question

Why does reaction rate increase dramatically with temperature?

For many reactions, a 10 °C rise can double or triple the rate. This cannot be explained by simple kinetic energy arguments alone — the relationship is exponential.

2. The Key Insight: An Energy Barrier

Before molecules can react, they must collide — but not all collisions lead to products.

There is a minimum energy threshold called activation energy ( $E_a$ ). Only collisions with energy ≥  $E_a$ can break old bonds and form new ones.

Think of it like pushing a boulder over a hill:

The hilltop is the transition state (activated complex).
$E_a$ is the height of that hill from the reactant valley.

3. The Boltzmann Factor — The "Why" of Exponential Dependence

At temperature $T$ , the fraction of molecules with energy ≥  $E_a$ is given by the Boltzmann distribution:

\text{Fraction} = e^{-E_a / (RT)}

where:

$R$ = universal gas constant (8.314 J mol⁻¹ K⁻¹)
$T$ = absolute temperature (Kelvin)

Why this form?

The Boltzmann distribution tells us that the probability of a molecule having energy $E$ is proportional to $e^{-E/(k_B T)}$ .
For 1 mole of molecules, $k_B$ (Boltzmann constant) becomes $R$ via $R = N_A k_B$ .
So the fraction with energy ≥ $E_a$ is the integral of that distribution from $E_a$ to ∞, which yields $e^{-E_a/(RT)}$ .

Key takeaway: This exponential factor is not arbitrary — it comes directly from statistical mechanics.

4. The Pre-Exponential Factor ( $A$ )

Even if a collision has enough energy, it must also:

Have the correct orientation (steric factor)
Occur with sufficient collision frequency

These are bundled into the pre-exponential factor $A$ (also called the frequency factor):

A = \text{(collision frequency)} \times \text{(orientation factor)}

For simple gas-phase reactions, collision frequency can be calculated from kinetic molecular theory — it's on the order of $10^{10}$  L mol⁻¹ s⁻¹.

5. Putting It Together: The Arrhenius Equation

The rate constant $k$ is proportional to:

The number of effective collisions per second (given by $A$ )
The fraction of collisions with sufficient energy (given by $e^{-E_a/(RT)}$ )

Thus:

\boxed{k = A \, e^{-E_a / (RT)}}

This is the Arrhenius equation.

6. Why It Works — The Physical Logic

Component	Physical meaning	Why it's there
$A$	Maximum possible rate if every collision worked	Accounts for collision frequency & geometry

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The Arrhenius equation gives the activation energy directly from the pre-exponential factor and the exponential term. Comparing the given form $k = A e^{-E_a/(RT)}$ with $k = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}$ shows that $E_a/R = 28000\ \text{K}$ , so $E_a = 28000 \times 8.314 \approx 2.33 \times 10^5\ \text{J/mol}$ or $233\ \text{kJ/mol}$ .

The Arrhenius equation is the backbone of chemical kinetics when it comes to temperature dependence. It tells us that the rate constant $k$ depends on two things: how often molecules collide with the right orientation (the pre-exponential factor $A$ ) and what fraction of those collisions have enough energy to overcome the barrier (the exponential term). That barrier is the activation energy $E_a$ .

The standard form is:

k = A e^{-E_a/(RT)}

where $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$ is the gas constant, and $T$ is the absolute temperature in Kelvin.

Now look at the given equation:

k = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}

The structure matches perfectly — the pre-exponential factor is $4.5\times10^{11}\ \text{s}^{-1}$ , and the exponent is $-28000/T$ . But in the standard form, the exponent is $-E_a/(RT)$ . So we just equate the two exponents.

Match the exponents The exponent in the given equation is $-\frac{28000}{T}$ . The exponent in the Arrhenius equation is $-\frac{E_a}{RT}$ . Since the temperature $T$ is in the denominator in both, we set:

\frac{E_a}{R} = 28000\ \text{K}

Solve for $E_a$ Multiply both sides by $R$ :

E_a = 28000\ \text{K} \times 8.314\ \text{J mol}^{-1}\text{K}^{-1}

Calculate $E_a = 28000 \times 8.314 = 232,792\ \text{J/mol}$ …

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Common Mistakes with Arrhenius Equation ( $k = A e^{-E_a/RT}$ )

Students often slip on this problem because the given equation looks almost like the standard form — but isn't. Here are the most frequent errors and how to avoid each.

✗ Mistake 1: Forgetting the gas constant $R$

The error:

Students see $e^{-28000\ K/T}$ and directly write $E_a = 28000\ \text{J/mol}$ or $28\ \text{kJ/mol}$ .

Why it's wrong:

The exponent in the standard Arrhenius form is $-\frac{E_a}{RT}$ , not $-\frac{E_a}{T}$ . The given $28000\ K$ is actually $\frac{E_a}{R}$ , not $E_a$ itself.

✓ How to avoid:

Always compare the given equation term-by-term with the standard form:

k = A e^{-E_a/RT}

Here:

k = (4.5 \times 10^{11}) e^{-28000/T}

So:

\frac{E_a}{R} = 28000\ \text{K}

Then solve:

E_a = 28000 \times R = 28000 \times 8.314 = 2.33 \times 10^5\ \text{J/mol}

Final answer:

E_a = \mathbf{2.33 \times 10^5\ J/mol\ (or\ 233\ kJ/mol)}

✗ Mistake 2: Using wrong value of $R$

The error:

Using $R = 0.0821\ \text{L·atm/mol·K}$ or $R = 8.314\ \text{J/mol·K}$ incorrectly.

Why it's wrong:

The units must match. Since $E_a$ is in J/mol, $R$ must be in J/mol·K.

✓ How to avoid:

Always use $R = \mathbf{8.314\ J\ mol^{-1} K^{-1}}$ for energy calculations. The 0.0821 value is only for PV = nRT with pressure in atm.

✗ Mistake 3: Misreading the pre-exponential factor units

The error:

Ignoring that $k$ has units of s $^{-1}$ (first-order reaction) and assuming it doesn't matter.

Why it's wrong:

While it doesn't affect $E_a$ calculation here, it's a common exam trap — they might ask for the order or the value of $A$ .

✓ How to avoid:

Note that $k$ has units of s $^{-1}$ → first-order reaction. The pre-exponential factor $A = 4.5 \times 10^{11}\ \text{s}^{-1}$ .

✗ Mistake 4: Confusing temperature dependence

The error:

Thinking $28000/T$ means $E_a$ changes with temperature.

Why it's wrong:

$E_a$ is constant for a given reaction. The $1/T$ dependence is in the exponent — $E_a$ itself is fixed. …

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Q.The decomposition of hydrocarbon follows the equation
$k = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}$
Calculate $E_a$ .

Concept understanding — Arrhenius Equation

The Arrhenius Equation Plot: Why Temperature Changes Reaction Speed

The core idea: an energy barrier

The Arrhenius equation (precise statement)

Why plot it? The linear trick

What the plot tells you

A typical Arrhenius plot looks like this

Why this formula?

Arrhenius Equation: Why It Holds

1. The Core Question

2. The Key Insight: An Energy Barrier

3. The Boltzmann Factor — The "Why" of Exponential Dependence

4. The Pre-Exponential Factor ( $A$ )

5. Putting It Together: The Arrhenius Equation

6. Why It Works — The Physical Logic

Method: Arrhenius Equation — Activation Energy from Exponential Form

Step 1: Compare the given equation with the standard form

Step 2: Equate the exponents

Step 3: Solve for $E_a$

Common Mistakes with Arrhenius Equation ( $k = A e^{-E_a/RT}$ )

✗ Mistake 1: Forgetting the gas constant $R$

✗ Mistake 2: Using wrong value of $R$

✗ Mistake 3: Misreading the pre-exponential factor units

✗ Mistake 4: Confusing temperature dependence

Q.The decomposition of hydrocarbon follows the equation k=(4.5×1011 s−1) e−28000 K/Tk = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}k=(4.5×1011 s−1)e−28000 K/T Calculate EaE_aEa​.

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Q.The decomposition of hydrocarbon follows the equation
$k = (4.5\times10^{11}\ \text{s}^{-1})\,e^{-28000\ K/T}$
Calculate $E_a$ .

Unlock everything free for 14 days