The rate constant for the decomposition of N2O5 at various temperatures is given below:
| T/°C | 0 | 20 | 40 | 60 | 80 |
|---|---|---|---|---|---|
| 105×k/s−1 | 0.0787 | 1.70 | 25.7 | 178 | 2140 |
Draw a graph between lnk and 1/T and calculate the values of A and Ea. Predict the rate constant at 30° and 50°C.
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🔒 Start your 14-day free trial to unlock the full solution →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 (Ea) — 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 ≥Ea is given by the Boltzmann distribution:
fraction=e−Ea/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=Ae−Ea/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)
- Ea = activation energy (J/mol or kJ/mol)
- R = 8.314 J/(mol·K)
- T = temperature in Kelvin
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=Ae−Ea/RT is exponential in 1/T. That's hard to eyeball. But take the natural logarithm of both sides:
lnk=lnA−REa⋅T1
This is the equation of a straight line:
y=c+mx
where:
- y=lnk
- x=1/T
- slope m=−Ea/R
- intercept c=lnA
So if you measure k at several temperatures and plot lnk versus 1/T, you get a straight line — provided the reaction follows Arrhenius behaviour (most do, over moderate temperature ranges).
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 Ea:
Ea=−(slope)×R
A steep negative slope means a large Ea — the reaction is very sensitive to temperature. A shallow slope means a small Ea — temperature doesn't affect it much.
From the intercept, you get A:
A=eintercept
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) | lnk |
|---|---|---|---|
| 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 lnk (y-axis) vs 1/T (x-axis). The points fall on a straight line. Draw the best-fit line, measure its slope, and compute Ea. …
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 (Ea). Only collisions with energy ≥ Ea 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).
- Ea 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 ≥ Ea is given by the Boltzmann distribution:
Fraction=e−Ea/(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/(kBT).
- For 1 mole of molecules, kB (Boltzmann constant) becomes R via R=NAkB.
- So the fraction with energy ≥ Ea is the integral of that distribution from Ea to ∞, which yields e−Ea/(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=(collision frequency)×(orientation factor)
For simple gas-phase reactions, collision frequency can be calculated from kinetic molecular theory — it's on the order of 1010 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−Ea/(RT))
Thus:
k=Ae−Ea/(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 |
The key idea is the Arrhenius equation in its linear form:
lnk=lnA−REa⋅T1.
A plot of lnk vs 1/T gives a straight line with slope =−Ea/R and intercept =lnA.
Step 1 – Convert data
Convert T to Kelvin: T/K=t/°C+273.15. The given k values are 105×k, so the actual k used for lnk is the table value ×10−5.
| T/°C | T/K | 103/T (K−1) | actual k (s−1) | lnk |
|---|---|---|---|---|
| 0 | 273.15 | 3.661 | 7.87×10−7 | −14.055 |
| 20 | 293.15 | 3.411 | 1.70×10−5 | −10.982 |
| 40 | 313.15 | 3.193 | 2.57×10−4 | −8.266 |
| 60 | 333.15 | 3.002 | 1.78×10−3 | −6.331 |
| 80 | 353.15 | 2.832 | 2.14×10−2 | −3.844 |
Step 2 – Plot and find slope
Plot lnk (y-axis) vs 103/T (x-axis). Using the first and last points:
Slope =(2.832−3.661)×10−3(−3.844)−(−14.055)=−0.829×10−310.211≈−1.232×104 K.
Step 3 – Calculate Ea and A
Ea=−slope×R=1.232×104×8.314≈1.024×105 J/mol = 102.4 kJ/mol.
Intercept lnA=lnk+REa⋅T1. Using the (central) point at 40°C:
lnA=−8.266+(1.232×104)(3.193×10−3)=−8.266+39.34=31.07
A=e31.07≈3.1×1013 s−1. …
Using the Arrhenius equation lnk=lnA−REa⋅T1, we plot lnk vs 1/T to get a straight line. From its slope (−Ea/R) and intercept (lnA), we find Ea≈102.4 kJ mol−1 and A≈3.1×1013 s−1. Then we predict k30∘C≈7.0×10−5 s−1 and k50∘C≈8.6×10−4 s−1.
The Arrhenius equation is the backbone of temperature-dependent kinetics. It tells us that the rate constant k depends exponentially on temperature:
k=Ae−Ea/RT
Taking natural logs gives a linear form:
lnk=lnA−REa⋅T1
This is of the form y=mx+c, where y=lnk, x=1/T, slope m=−Ea/R, and intercept c=lnA. So if we plot lnk against 1/T, we get a straight line — and from its slope and intercept we can extract both Ea and A.
Temperature must be in kelvin when using 1/T in the Arrhenius plot. A common mistake is to use Celsius directly — that gives a completely wrong slope.
Let’s work through it step by step.
1. Convert temperatures to kelvin and compute 1/T and lnk
| T (°C) | T (K) | 1/T (K−1) | k (s−1) | lnk |
|---|---|---|---|---|
| 0 | 273.15 | 3.661×10−3 | 0.0787×10−5 | −14.055 |
| 20 | 293.15 | 3.411×10−3 | 1.70×10−5 | −10.982 |
| 40 | 313.15 | 3.193×10−3 | 25.7×10−5 | −8.266 |
| 60 | 333.15 | 3.002×10−3 | 178×10−5 | −6.331 |
| 80 | 353.15 | 2.832×10−3 | 2140×10−5 | −3.844 |
Notice that k values are given as 105×k, so we divide by 105 to get actual k in s−1.
2. Plot lnk vs 1/T
On a graph, the points fall beautifully on a straight line. The slope is negative (since k increases with T, lnk increases as 1/T decreases). We can calculate the slope using any two well-separated points, but for accuracy, use the first and last:
slope=Δ(1/T)Δ(lnk)=(2.832−3.661)×10−3(−3.844)−(−14.055)=−0.829×10−310.211≈−1.232×104 K
Using the two extreme points gives a quick estimate. For exam problems, this is usually sufficient — but if you have time, a least-squares fit (or averaging slopes from multiple pairs) gives a more reliable result.
3. Calculate Ea from the slope
Since slope =−Ea/R, we have:
−REa=−1.232×104 K
Ea=1.232×104×R=1.232×104×8.314 J mol−1
Ea≈1.024×105 J mol−1=102.4 kJ mol−1
4. Calculate A from the intercept
The intercept c=lnA. From the graph, the line crosses the lnk axis at 1/T=0 (theoretical). Using the point-slope form with any data point, say at T=40∘C:
lnA=lnk+REa⋅T1
lnA=−8.266+(1.232×104)×(3.193×10−3)
lnA=−8.266+39.34=31.07
So:
A=e31.07≈3.1×1013 s−1 …
Method: Graphical Arrhenius Analysis (Two-Point & Linear Regression)
The Arrhenius equation in logarithmic form is:
lnk=lnA−REa⋅T1
This is a straight line: y=mx+c, where:
- y=lnk
- x=1/T (in Kelvin)
- Slope m=−Ea/R
- Intercept c=lnA
Step 1: Convert temperatures to Kelvin and compute 1/T and lnk
The given k values in the table are 105×k, so the actual k used to compute lnk is the table value ×10−5.
| T (°C) | T (K) | 1/T (K−1) | k×105 (s−1) | actual k (s−1) | lnk |
|---|---|---|---|---|---|
| 0 | 273.15 | 3.661×10−3 | 0.0787 | 7.87×10−7 | −14.055 |
| 20 | 293.15 | 3.411×10−3 | 1.70 | 1.70×10−5 | −10.982 |
| 40 | 313.15 | 3.193×10−3 | 25.7 | 2.57×10−4 | −8.266 |
| 60 | 333.15 | 3.002×10−3 | 178 | 1.78×10−3 | −6.331 |
| 80 | 353.15 | 2.832×10−3 | 2140 | 2.14×10−2 | −3.844 |
Step 2: Plot lnk (y-axis) vs 1/T (x-axis)
You will get a straight line with a negative slope.
Step 3: Calculate slope from the graph
Using the first and last points (for a quick estimate):
slope=Δ(1/T)Δlnk=(2.832−3.661)×10−3−3.844−(−14.055)
=−0.829×10−310.211≈−1.232×104 K
Step 4: Calculate activation energy Ea
From slope m=−Ea/R:
Ea=−m×R=1.232×104×8.314
Ea≈1.024×105 J/mol=102.4 kJ/mol
Step 5: Calculate pre-exponential factor A
From intercept c=lnA:
Using the central point at T=313.15 K (1/T=3.193×10−3, lnk=−8.266):
lnA=lnk+REa⋅T1
lnA=−8.266+(1.232×104)(3.193×10−3)
lnA=−8.266+39.34=31.07
A=e31.07≈3.1×1013 s−1
Step 6: Predict rate constants at 30°C and 50°C
Use the fitted line lnk=31.07−T1.232×104.
At 30°C (303.15 K):
lnk=31.07−(1.232×104)(3.299×10−3) …
Common Mistakes & How to Avoid Them (Arrhenius Equation)
1. ✗ Forgetting to convert temperature to Kelvin
The Mistake: Students plot 1/T using °C values directly (e.g., 1/20 instead of 1/293).
Why it's wrong: The Arrhenius equation uses absolute temperature:
k=Ae−Ea/RT
T must be in Kelvin (K=°C+273).
✓ How to avoid: Always write the conversion step explicitly:
- 0°C=273K
- 20°C=293K
- 40°C=313K, etc.
2. ✗ Using k directly instead of lnk
The Mistake: Plotting k vs 1/T (a curve) instead of lnk vs 1/T (a straight line).
Why it's wrong: The Arrhenius equation in linear form is:
lnk=lnA−REa⋅T1
Only lnk vs 1/T gives a straight line with slope =−Ea/R.
✓ How to avoid: Before plotting, compute lnk for each k value. Use a table:
| T/K | 105k | lnk | 1/T |
|---|---|---|---|
| 273 | 0.0787 | ln(0.0787×10−5) | 0.00366 |
| ... | ... | ... | ... |
3. ✗ Mishandling the 105 factor in k
The Mistake: Taking ln(0.0787) instead of ln(0.0787×10−5).
Why it's wrong: The given k values are 105×k. So actual k=(table value)×10−5.
✓ How to avoid: Write clearly:
kactual=(table value)×10−5
Then take ln of this actual value.
4. ✗ Using R in wrong units
The Mistake: Using R=0.0821 (L·atm/mol·K) instead of R=8.314 (J/mol·K).
Why it's wrong: Ea is typically in J/mol or kJ/mol. The correct R for energy calculations is:
R=8.314 J mol−1K−1
✓ How to avoid: Remember:
- For Ea in J/mol: use R=8.314
- For Ea in kJ/mol: use R=0.008314
5. ✗ Confusing slope sign when finding Ea
The Mistake: Taking Ea=slope×R instead of Ea=−slope×R.
Why it's wrong: From lnk=lnA−REa⋅T1, the slope is negative:
slope=−REa
So Ea=−slope×R (which gives a positive value).
✓ How to avoid:
- Plot the graph
- Calculate slope =Δ(1/T)Δlnk (will be negative)
- Then Ea=−slope×R
6. ✗ Using wrong points for interpolation at 30°C and 50°C
The Mistake: Reading k directly from the curved k vs T plot.
Why it's wrong: The relationship is linear only for lnk vs 1/T. …
- TG EAPCET 2024Set eng-2024-05-11-FN1 markMCQQ.At what temperature will the RMS velocity of sulphur dioxide molecules at 400 K be the same as the most probable velocity of oxygen molecules? (A) 600 K (B) 200 K (C) 400 K (D) 300 K
›Reveal solutionSolution
Equating vrms(SO2,400K) with vmp(O2,T) gives T=300 K.
vrms=M13RT1,vmp=M22RT2.
With M(SO2)=64, T1=400 K, and M(O2)=32, set the two speeds equal:
M13RT1=M22RT2 …
- TG EAPCET 2021Set ap-2021-08-10-AN1 markMCQQ.The rate constant of a reaction at 25 ∘C is 1×10−3 min−1. If the temperature coefficient of the reaction is 2, the rate constant (min−1) at 15 ∘C is (A) 2×10−3 (B) 2×10−4 (C) 5×10−4 (D) 4×10−3
›Reveal solutionSolution
The temperature coefficient (Q₁₀ = 2) tells us the rate constant halves for every 10 °C drop. From 25 °C to 15 °C is a 10 °C decrease, so the rate constant at 15 °C is half of 1×10−3 min⁻¹, i.e., 5×10−4 min⁻¹.
Concept & Intuition
The temperature coefficient (often denoted Q10) is defined as the factor by which the rate constant increases when the temperature is raised by 10 °C.
Here Q10=2 means:
- Increase temperature by 10 °C → rate constant doubles.
- Decrease temperature by 10 °C → rate constant halves.
We are moving down from 25 °C to 15 °C, a drop of exactly 10 °C. So the rate constant at the lower temperature is simply k15=k25/2.
Step-by-step reasoning
-
Identify the given data
- k25=1×10−3 min⁻¹
- Temperature coefficient Q10=2
- Temperature change: from 25 °C to 15 °C = −10 °C
-
Apply the definition of temperature coefficient
The relationship is:
kTkT+10=Q10
So for a 10 °C decrease:
k25k15=Q101=21
- Calculate the unknown rate constant k15=k25×21=(1×10−3)×21=0.5×10−3=5×10−4 min−1 …
- TG EAPCET 2021Set ap-2021-08-10-FN1 markMCQQ.The rate constant of a reaction at 25∘C is 1×10−3 min−1. If the temperature coefficient of the reaction is 2, the rate constant (min−1) at 15∘C is (A) 2×10−3 (B) 2×10−4 (C) 5×10−4 (D) 4×10−3
›Reveal solutionSolution
The temperature coefficient (Q₁₀ = 2) tells us the rate constant halves for every 10°C drop. Going from 25°C to 15°C is a 10°C decrease, so the rate constant at 15°C is half of the given value: 5×10−4 min−1.
Concept & Intuition
The temperature coefficient (often denoted Q10) is the factor by which the rate constant changes when the temperature is raised by 10°C. Here Q10=2 means that for every 10°C increase, the rate constant doubles; conversely, for every 10°C decrease, it halves. Since we are moving from 25°C down to 15°C (a drop of exactly 10°C), we simply divide the given rate constant by 2.
Step-by-step reasoning
-
Identify the temperature change
The given rate constant k25=1×10−3 min−1 is at 25∘C. We need k15 at 15∘C.
The difference: 25∘C−15∘C=10∘C.
-
Apply the definition of the temperature coefficient
The temperature coefficient Q10 is defined as
Q10=kTkT+10
Here Q10=2, so for a 10°C increase the rate constant multiplies by 2. For a 10°C decrease, we use the reciprocal:
-
- TG EAPCET 2021Set eng-2021-08-05-AN1 markMCQQ.When salt is added to water, which of the following statement is true? (A) Boiling point decreases (B) Boiling point increases (C) Boiling point remain constant (D) Freezing point increases
›Reveal solutionSolution
Adding salt to water raises its boiling point and lowers its freezing point — a colligative effect. The correct statement is that the boiling point increases.
The question tests your understanding of colligative properties — properties that depend only on the number of solute particles, not their chemical identity. When salt (sodium chloride, NaCl) dissolves in water, it dissociates into Na⁺ and Cl⁻ ions, increasing the total number of particles in the solution. This changes two key physical properties of water: its boiling point and its freezing point.
Why does this happen? At the boiling point, the vapour pressure of the liquid equals the atmospheric pressure. Adding a non-volatile solute like salt lowers the vapour pressure of the solvent (water). To make the vapour pressure reach atmospheric pressure again, you need to supply more heat — hence the boiling point rises. For freezing, the solute particles disrupt the orderly arrangement of water molecules into ice, so a lower temperature is needed to freeze the solution — hence the freezing point drops.
Let’s examine each option step by step.
-
Boiling point behaviour
The boiling point elevation is given by ΔTb=i⋅Kb⋅m, where i is the van’t Hoff factor (for NaCl, i≈2), Kb is the ebullioscopic constant of water, and m is the molality. Since ΔTb>0, the boiling point increases. This eliminates option (A) and (C).
-
Freezing point behaviour …
-
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