The rate constant for the decomposition of hydrocarbons is $2.418\times10^{-5}\ \text{s}^{-1}$ at 546 K. If the energy of activation is $179.9\ \text{kJ/mol}$, what will be the value of pre-exponential factor?

> [!TLDR] > Using the Arrhenius equation $k = A e^{-E_a/(RT)}$, we solve for the pre-exponential factor $A$. With $k = 2.418 \times 10^{-5}\ \text{s}^{-1}$, $E_a = 179.9\ \text{kJ/mol}$, and $T = 546\ \text{K}$, the value of $A$ is approximately $3.9 \times 10^{12}\ \text{s}^{-1}$. The Arrhenius equation is the backbone of chemical kinetics when temperature dependence is involved. It tells us that the rate constant $k$ depends on two things: the fraction of molecules with enough energy to react (given by the exponential term $e^{-E_a/(RT)}$) and how often collisions happen in the right orientation (the pre-exponential factor $A$). Here, we know $k$, $E_a$, and $T$, and we need to find $A$ — so we simply rearrange the equation. > [!FORMULA] > $$k = A e^{-E_a/(RT)}$$ Where: - $k$ = rate constant ($\text{s}^{-1}$) - $A$ = pre-exponential factor (same units as $k$) - $E_a$ = activation energy (J/mol — careful with units!) - $R$ = gas constant ($8.314\ \text{J mol}^{-1}\ \text{K}^{-1}$) - $T$ = temperature (K) 1. **Convert activation energy to consistent units.** The given $E_a = 179.9\ \text{kJ/mol}$. Since $R$ is in J/mol·K, we must convert: $$E_a = 179.9 \times 10^3\ \text{J/mol} = 1.799 \times 10^5\ \text{J/mol}$$ 2. **Compute the exponent $\frac{E_a}{RT}$.** First, $RT = 8.314 \times 546$. $$RT = 8.314 \times 546 = 4539.444\ \text{J/mol}$$ Then, $$\frac{E_a}{RT} = \frac{1.799 \times 10^5}{4539.444} \approx 39.63$$ 3. **Find $e^{-E_a/(RT)}$.** $$e^{-39.63} = \frac{1}{e^{39.63}}$$ This is a very small number. Using a calculator: $$e^{39.63} \approx 1.62 \times 10^{17}$$ So, $$e^{-39.63} \approx 6.17 \times 10^{-18}$$ 4. **Rearrange the Arrhenius equation to solve for $A$.** $$A = \frac{k}{e^{-E_a/(RT)}} = k \times e^{E_a/(RT)}$$ Substitute the values: $$A = (2.418 \times 10^{-5}) \times (1.62 \times 10^{17})$$ 5. **Calculate $A$.** $$A = 2.418 \times 1.62 \times 10^{12} \approx 3.917 \t

The rate constant for the decomposition of hydrocarbons is 2.418×10⁻⁵s⁻¹ at 546 K. If the energy of activation

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2.418\times10^{-5}\ \text{s}^{-1}

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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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Using the Arrhenius equation $k = A e^{-E_a/(RT)}$ , we solve for the pre-exponential factor $A$ . With $k = 2.418 \times 10^{-5}\ \text{s}^{-1}$ , $E_a = 179.9\ \text{kJ/mol}$ , and $T = 546\ \text{K}$ , the value of $A$ is approximately $3.9 \times 10^{12}\ \text{s}^{-1}$ .

The Arrhenius equation is the backbone of chemical kinetics when temperature dependence is involved. It tells us that the rate constant $k$ depends on two things: the fraction of molecules with enough energy to react (given by the exponential term $e^{-E_a/(RT)}$ ) and how often collisions happen in the right orientation (the pre-exponential factor $A$ ). Here, we know $k$ , $E_a$ , and $T$ , and we need to find $A$ — so we simply rearrange the equation.

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

Where:

$k$ = rate constant ( $\text{s}^{-1}$ )
$A$ = pre-exponential factor (same units as $k$ )
$E_a$ = activation energy (J/mol — careful with units!)
$R$ = gas constant ( $8.314\ \text{J mol}^{-1}\ \text{K}^{-1}$ )
$T$ = temperature (K)

Convert activation energy to consistent units. The given $E_a = 179.9\ \text{kJ/mol}$ . Since $R$ is in J/mol·K, we must convert:

E_a = 179.9 \times 10^3\ \text{J/mol} = 1.799 \times 10^5\ \text{J/mol}

Compute the exponent $\frac{E_a}{RT}$ . First, $RT = 8.314 \times 546$ .

RT = 8.314 \times 546 = 4539.444\ \text{J/mol}

Then,

\frac{E_a}{RT} = \frac{1.799 \times 10^5}{4539.444} \approx 39.63

Find $e^{-E_a/(RT)}$ .

e^{-39.63} = \frac{1}{e^{39.63}}

This is a very small number. Using a calculator:

e^{39.63} \approx 1.62 \times 10^{17}

So,

e^{-39.63} \approx 6.17 \times 10^{-18}

Rearrange the Arrhenius equation to solve for $A$ .

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

Substitute the values:

A = (2.418 \times 10^{-5}) \times (1.62 \times 10^{17})

Calculate $A$ . $A = 2.418 \times 1.62 \times 10^{12} \approx 3.917 \times 10^{12}\ \text{s}^{-1}$ …

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Common Mistakes with the Arrhenius Equation (Pre-exponential Factor)

This is a classic "plug-and-chug" problem from Chemical Kinetics. While the calculation is straightforward, students often lose marks due to unit errors and logarithm mishandling.

Mistake #1: Forgetting to Convert Activation Energy to J/mol

The error:

The activation energy $E_a$ is given in kJ/mol (179.9 kJ/mol), but the gas constant $R$ is usually taken as $8.314\ \text{J mol}^{-1}\text{K}^{-1}$ . If you plug in $E_a$ in kJ without converting, the ratio $\frac{E_a}{RT}$ becomes 1000 times too small, giving a wildly wrong $A$ .

How to avoid:

Always convert $E_a$ to J/mol before using it in the Arrhenius equation.

E_a = 179.9 \times 10^3\ \text{J/mol}

Quick check: If your $\frac{E_a}{RT}$ value is less than about 10 (for typical exam problems), you likely forgot the conversion.

Mistake #2: Using the Wrong Form of the Arrhenius Equation

The error:

Students sometimes use the integrated form (for finding $k$ at two temperatures) when the problem only gives one temperature and asks for $A$ .

How to avoid:

Recognize the direct form:

k = A e^{-E_a / RT}

Here, you have $k$ , $E_a$ , and $T$ — solve directly for $A$ :

A = k \cdot e^{E_a / RT}

No need for two temperatures or the logarithmic form unless the problem asks for it.

Mistake #3: Mishandling the Natural Logarithm When Solving

The error:

When rearranging $k = A e^{-E_a/RT}$ , some students incorrectly write:

A = \frac{k}{e^{-E_a/RT}} \quad \text{(correct)}

but then compute $e^{-E_a/RT}$ instead of $e^{+E_a/RT}$ .

How to avoid:

Rewrite clearly:

A = k \times e^{+E_a/RT}

Remember: The negative sign in the exponent flips when you bring $e^{-E_a/RT}$ to the other side.

Mistake #4: Incorrectly Computing the Exponential

The error:

Students often compute $e^{E_a/RT}$ by first calculating $\frac{E_a}{RT}$ and then pressing the $e^x$ button — but they forget to use parentheses on their calculator, leading to:

e^{179900} / (8.314 \times 546) \quad \text{(wrong!)}

instead of:

e^{179900 / (8.314 \times 546)} \quad \text{(correct)}

How to avoid:

On a scientific calculator, type: …

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Q.The rate constant for the decomposition of hydrocarbons is $2.418\times10^{-5}\ \text{s}^{-1}$ at 546 K. If the energy of activation is $179.9\ \text{kJ/mol}$ , what will be the value of pre-exponential factor?

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: Two-Point Arrhenius Form (Solving for Pre-Exponential Factor)

Steps

Common Mistakes with the Arrhenius Equation (Pre-exponential Factor)

Mistake #1: Forgetting to Convert Activation Energy to J/mol

Mistake #2: Using the Wrong Form of the Arrhenius Equation

Mistake #3: Mishandling the Natural Logarithm When Solving

Mistake #4: Incorrectly Computing the Exponential

Q.The rate constant for the decomposition of hydrocarbons is 2.418×10−5 s−12.418\times10^{-5}\ \text{s}^{-1}2.418×10−5 s−1 at 546 K. If the energy of activation is 179.9 kJ/mol179.9\ \text{kJ/mol}179.9 kJ/mol, what will be the value of pre-exponential factor?

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Q.The rate constant for the decomposition of hydrocarbons is $2.418\times10^{-5}\ \text{s}^{-1}$ at 546 K. If the energy of activation is $179.9\ \text{kJ/mol}$ , what will be the value of pre-exponential factor?

Unlock everything free for 14 days