For the decomposition of azoisopropane to hexane and nitrogen at 543 K, the following data are obtained.
| t (sec) | P (mm of Hg) |
|---|---|
| 0 | 35.0 |
| 360 | 54.0 |
| 720 | 63.0 |
Calculate the rate constant.
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — First Order Kinetics
First Order Kinetics
Imagine you have a bucket of water with a small hole at the bottom. The water drains out. At the start, the bucket is full, so the pressure at the hole is high — water gushes out fast. As the water level drops, the pressure decreases, and the water trickles out more slowly. The rate at which water leaves is directly proportional to how much water is still in the bucket.
That's the core intuition behind first order kinetics: the rate of a process depends linearly on how much of the substance is left.
The Precise Statement
In chemistry, first order kinetics describes a reaction where the rate of the reaction is directly proportional to the concentration of one reactant.
If we have a reaction: A→products, then:
Rate=−dtd[A]=k[A]
Here:
- [A] is the concentration of reactant A at any time t
- k is the rate constant (units: time−1, e.g., s−1)
- The negative sign indicates that [A] decreases over time
The key point: double the concentration, double the rate. Halve the concentration, halve the rate.
The Integrated Form — What Actually Happens Over Time
The differential equation above tells us the instantaneous rate. But what we usually want is: how does concentration change with time?
Integrating gives:
ln[A]t=ln[A]0−kt
or equivalently:
[A]t=[A]0e−kt
Where [A]0 is the initial concentration and [A]t is the concentration at time t.
This exponential decay is the hallmark of first order kinetics. The concentration drops rapidly at first, then more slowly, approaching zero asymptotically.
The Half-Life — A Beautiful Constant
For first order kinetics, the half-life (t1/2) — the time taken for half the reactant to be consumed — is independent of the starting concentration.
t1/2=kln2≈k0.693
This is a powerful result. Whether you start with 100 g or 1 g, it always takes the same time to go from that amount to half of it. This is unique to first order kinetics — no other order has this property.
For a first order process, after n half-lives, the fraction remaining is (21)n. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. And so on.
How to Identify First Order Kinetics Experimentally
If you plot ln[A] versus time and get a straight line with slope −k, the reaction is first order. This is the gold standard test.
Alternatively, if the half-life remains constant as you change the initial concentration, that's a strong indicator.
Real-World Examples
- Radioactive decay: Every radioactive isotope decays by first order kinetics. Carbon-14 dating works because t1/2=5730 years, regardless of how much carbon-14 is present. …
Why this formula?
First Order Kinetics: Why the Formula Holds
Let's build this from the core idea — not just memorise the equation.
The Fundamental Assumption
In a first order reaction, the rate of reaction depends linearly on the concentration of only one reactant.
If we have:
A→products
The rate law is:
Rate=−dtd[A]=k[A]
Here:
- −dtd[A] = rate of disappearance of A (negative because [A] decreases)
- k = rate constant (units: time−1, e.g., s−1)
- [A] = concentration of A at any time
Why linear? Because the probability of a single molecule reacting in a given time is constant — it doesn't depend on other molecules. This is the molecular logic behind first order.
Deriving the Integrated Rate Law
We start from the differential form:
−dtd[A]=k[A]
Step 1: Separate variables
Bring all [A] terms to one side, dt to the other:
[A]d[A]=−kdt
Step 2: Integrate both sides
Integrate from initial time t=0 (concentration [A]0) to any time t (concentration [A]t):
∫[A]0[A]t[A]d[A]=−k∫0tdt
The left side integrates to ln[A]:
ln[A]t−ln[A]0=−kt
Step 3: Rearrange
ln[A]0[A]t=−kt
Or equivalently:
ln[A]t=ln[A]0−kt
This is the integrated rate law for first order kinetics.
Why This Form Makes Sense
- Exponential decay: Taking antilog:
[A]t=[A]0e−kt
The concentration decays exponentially — a hallmark of first order processes.
- Constant half-life: The time for [A]t to become half of [A]0 is:
2[A]0=[A]0e−kt1/2
21=e−kt1/2
ln(21)=−kt1/2
t1/2=kln2
Key insight: t1/2 is independent of initial concentration — unique to first order. This is why radioactive decay (a first order process) has a fixed half-life regardless of how much you start with.
Graphical Interpretation (Exam-Ready) …
Concept: Average Rate of Reaction – For a gaseous reaction, the rate constant can be found using the integrated rate law for a first-order process, relating total pressure to the extent of reaction.
The reaction is:
(CH3)2CHN=NCH(CH3)2→C6H14+N2
Let initial pressure of azoisopropane be P0=35.0 mm Hg. At time t, let x be the decrease in pressure of reactant. Then:
- Pressure of azoisopropane = P0−x
- Pressure of hexane = x
- Pressure of nitrogen = x
Total pressure, Pt=(P0−x)+x+x=P0+x
So, x=Pt−P0
For a first-order gas-phase reaction:
k=t2.303logP0−xP0=t2.303log2P0−PtP0
At t=360 s:
k=3602.303log2(35.0)−54.035.0=3602.303log16.035.0
=3602.303×0.3399=2.17×10−3 s−1 (unrounded: 2.174×10−3)
At t=720 s: …
For a first-order gas-phase reaction, the rate constant can be found from the pressure increase over time. Using the formula k=t2.303log2P0−PtP0, the calculated value (average of the two time points) is 2.20×10−3 s−1.
The key insight here is that we are dealing with a gas-phase decomposition where the total pressure changes as the reaction proceeds. For the reaction:
(CH3)2CHN=NCH(CH3)2→C6H14+N2
azoisopropane decomposes to give hexane and nitrogen gas. Since all species are gases, the total pressure at any time is the sum of the partial pressures of the reactant and products.
Why does this matter? In a constant-volume container at fixed temperature, pressure is proportional to the number of moles. As one molecule of azoisopropane breaks into two molecules (one hexane + one nitrogen), the total number of moles increases. This means the total pressure rises over time — and that rise tells us exactly how much reactant has decomposed.
The reaction follows first-order kinetics (typical for such decompositions), so we can use the integrated rate law for a first-order reaction in terms of pressure.
Let’s work through it step by step.
- Define the initial and final pressures.
Let P0 be the initial pressure of azoisopropane alone. At t=0, P0=35.0 mm Hg.
Let Pt be the total pressure at time t.
If x is the decrease in pressure of azoisopropane at time t, then:
- Pressure of azoisopropane remaining = P0−x
- Pressure of hexane produced = x (since 1 mole gives 1 mole)
- Pressure of nitrogen produced = x So total pressure:
Pt=(P0−x)+x+x=P0+x
Therefore, x=Pt−P0.
- Express the concentration of reactant in terms of pressure. For a first-order reaction, the rate constant k is given by:
k=t2.303log[A]t[A]0
Since pressure is proportional to concentration (ideal gas law at constant T and V), we can write:
k=t2.303logP0−xP0
Substituting x=Pt−P0:
k=t2.303logP0−(Pt−P0)P0=t2.303log2P0−PtP0
k=t2.303log2P0−PtP0
- Calculate k using data at t=360 sec. P0=35.0, Pt=54.0
2P0−Pt=70.0−54.0=16.0
2P0−PtP0=16.035.0=2.1875
log(2.1875)≈0.3399
k=3602.303×0.3399=3600.7827≈2.174×10−3 s−1
- Calculate k using data at t=720 sec to verify consistency. Pt=63.0 …
Method: Integrated Rate Law for First-Order Gas-Phase Reaction (Using Pressure Data)
This is a first-order gas-phase reaction where the total pressure changes as the reaction proceeds. We use the relationship between total pressure and reactant pressure to apply the first-order integrated rate law.
Step 1: Write the reaction and stoichiometry
(CH3)2CHN=NCH(CH3)2→N2+C6H14
Let initial pressure of azoisopropane = P0=35.0 mm Hg at t=0.
At time t, let x be the decrease in pressure of azoisopropane. Then:
- Pressure of azoisopropane remaining = P0−x
- Pressure of N2 formed = x
- Pressure of C6H14 formed = x
Total pressure at time t:
Pt=(P0−x)+x+x=P0+x
So, x=Pt−P0
Step 2: Express reactant pressure in terms of total pressure
Pressure of azoisopropane at time t:
PA=P0−x=P0−(Pt−P0)=2P0−Pt
Step 3: Apply first-order integrated rate law
For a first-order reaction:
k=t2.303logPAP0
Substitute PA=2P0−Pt:
k=t2.303log2P0−PtP0
Step 4: Calculate k for each data point
At t=360 s:
k=3602.303log2(35.0)−54.035.0=3602.303log70.0−54.035.0
=3602.303log16.035.0=3602.303log(2.1875)
log(2.1875)≈0.3398
k=3602.303×0.3398=3600.7825≈2.17×10−3 s−1
At t=720 s: …
Common Mistakes & How to Avoid Them
Mistake 1: Confusing Total Pressure with Partial Pressure of Reactant
The Error:
Students directly plug the given total pressure (P) into the first-order rate equation:
k=t2.303logPtP0
This is wrong because P0 and Pt in the table are total pressures, not the partial pressure of azoisopropane.
Why it's wrong:
As the reaction proceeds, the number of gas moles increases (1 mole reactant → 2 moles products). Total pressure rises even as reactant decreases. Using total pressure directly gives a meaningless result.
How to Avoid:
Always ask: "Is this the pressure of the reactant or the total pressure of the mixture?"
For gas-phase reactions with mole change, you must convert total pressure into partial pressure of reactant using stoichiometry.
Mistake 2: Forgetting the Stoichiometric Mole Change
The Error:
Students assume Preactant=Ptotal at all times.
Why it's wrong:
The reaction is:
(CH3)2CHN=NCH(CH3)2→C6H14+N2
- Initial: 1 mole → 0 moles
- At time t: (1−x) moles reactant, x moles hexane, x moles nitrogen
- Total moles at time t = (1−x)+x+x=1+x
So total pressure Pt is not proportional to reactant remaining.
How to Avoid:
Write the balanced equation and count moles. Let:
- P0 = initial pressure of pure reactant
- Pt = total pressure at time t
- pA = partial pressure of reactant at time t
From stoichiometry:
Pt=pA+2(P0−pA)=2P0−pA
Therefore:
pA=2P0−Pt
Always derive this relation before plugging into the rate equation.
Mistake 3: Using the Wrong Order Formula
The Error:
Applying zero-order or second-order formulas without checking.
Why it's wrong:
For gas-phase decomposition with mole increase, first-order kinetics is standard unless stated otherwise. Using the wrong order gives a non-constant k.
How to Avoid:
- For decomposition reactions, assume first order unless data suggests otherwise.
- Verify by checking if k is constant across time intervals using:
k=t2.303logpAP0
Mistake 4: Arithmetic Errors in the pA Calculation
The Error:
Miscalculating 2P0−Pt or using P0 incorrectly.
Example of error:
At t=360, Pt=54.0, P0=35.0
Wrong: pA=2(35.0)−54.0=70−54=16 ✓ (correct) …
Showing the 12 most recent of 26 on this concept.
- CBSE 2026Set 56/1/11 markMCQQ.Which of the following curve represents the first order reaction ? (A) A graph of t1/2 (y-axis) against initial concentration [R]0 (x-axis): a straight line rising from the origin (B) A graph of t1/2 against [R]0: a horizontal straight line (t1/2 independent of [R]0) (C) A graph of Rate against Concentration: a horizontal straight line (D) A graph of Rate against Concentration: a curve that falls as concentration increases
›Reveal solutionSolution
For a first-order reaction, the half-life t1/2 is independent of the initial concentration [R]0, so the correct plot is a horizontal straight line on a t1/2 vs. [R]0 graph — option (B).
The key to this question is knowing how the half-life of a reaction depends on the initial concentration — and that dependence is different for different orders. Let’s build the intuition from the Arrhenius equation and the integrated rate laws.
Why this approach works
For a first-order reaction, the rate law is:
Rate=k[R]
where k is the rate constant. The integrated form gives:
ln[R][R]0=kt
The half-life t1/2 is the time when [R]=2[R]0. Substituting:
ln[R]0/2[R]0=kt1/2⇒ln2=kt1/2
So:
t1/2=kln2
Notice: no [R]0 appears in this expression. That’s the defining feature — for a first-order reaction, the half-life is a constant, determined only by the rate constant k.
Now let’s examine each option.
-
Option (A): A straight line rising from the origin on a t1/2 vs. [R]0 graph. This would mean t1/2∝[R]0, which is true for a zero-order reaction (where t1/2=[R]0/2k). Not first-order.
-
Option (B): A horizontal straight line — t1/2 does not change as [R]0 changes. This matches t1/2=ln2/k, a constant. This is the correct plot for a first-order reaction.
-
Option (C): A graph of Rate vs. Concentration that is a horizontal straight line. That would mean Rate is independent of concentration — which is true for a zero-order reaction (Rate = k). For first-order, Rate = k[R], so the plot is a straight line through the origin, not horizontal. …
-
- CBSE 2026Set A1 markMCQQ.Which of the following is not a first order reaction ?(a) CH3COOC2H5 + H2O --(H+)--> CH3COOH + C2H5OH(b) CH3COOC2H5 + NaOH --> CH3COONa + C2H5OH(c) 2H2O2 --> 2H2O + O2(d) 2N2O5 --> 4NO2 + O2
›Reveal solutionSolution
Ester hydrolysis by NaOH (saponification) is second order (first order in ester and first order in OH-), so it is NOT a first-order reaction.
- (a) Acid hydrolysis of ester with excess water is pseudo-first order. …
- CBSE 2026Set ANNUAL1 markMCQQ.Acid hydrolysis of ethyl acetate is:(a) Zero order reaction(b) First order reaction(c) Second order reaction(d) Third order reaction
›Reveal solutionSolution
Acid hydrolysis of ethyl acetate is a classic example of a pseudo first order reaction.
The reaction is: CH3COOC2H5+H2OH+CH3COOH+C2H5OH. Strictly, this reaction depends on the concentrations of BOTH the ester and water, and should be second order overall (first order in each). However, water is used as the solvent and is present in vast molar excess compared to the ester, so as the reaction proceeds its concentration barely changes and can be treated as effectively constant.
…
- CBSE 2026Set ANNUAL1 markMCQQ.The value of rate constant of a pseudo-first-order reaction(a) depends on the concentration of reactants present in small amount(b) depends on the concentration of reactants present in excess(c) is independent of the concentration of the reaction(d) depends only on temperature
›Reveal solutionSolution
A pseudo-first-order rate constant is not a true elementary-step constant — it already has the (essentially fixed) concentration of the reactant present in excess multiplied into it, so its numerical value depends on how much of that excess reactant was used.
Example — acid-catalysed hydrolysis of ethyl acetate:
CH3COOC2H5+H2OH+CH3COOH+C2H5OH
The true rate law is rate=k[ester][H2O]. Since water is the solvent and is present in huge excess, [H2O] stays essentially constant throughout the reaction, so rate=k′[ester] where k′=k[H2O]. Experimentally the reaction looks first order (only [ester] appears), but the measured k′ is really the true rate constant k multiplied by whatever fixed [H2O] happened to be present.
Why the other options are wrong:
- (a) It is the concentration of the reactant present in excess — not the one present in a small amount — that gets folded into kobs. …
- CBSE 2025Set 56/6/11 markMCQQ.For the following question, two statements are given — one labelled as Assertion (A) and the other labelled as Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true. Assertion (A) : Hydrolysis of an ester follows first order kinetics. Reason (R) : The concentration of water does not get altered much during the reaction.
›Reveal solutionSolution
The assertion is true because ester hydrolysis is pseudo-first order; the reason correctly explains why — water is in large excess so its concentration stays nearly constant, making the observed kinetics first order.
The Concept: Why First Order Kinetics Appears
When you study reaction kinetics, the order of a reaction tells you how the rate depends on the concentrations of reactants. For a true bimolecular reaction like ester hydrolysis:
CH3COOC2H5+H2OH+CH3COOH+C2H5OH
The rate law should be:
Rate=k[ester][H2O]
That would make it second order overall — first order in ester and first order in water. But here’s the twist: in practice, the reaction is carried out in aqueous solution where water is the solvent. Its concentration is about 55.5 M, while the ester concentration is typically 0.1 M or less. So water is in huge excess.
TipWhen one reactant is present in such large excess that its concentration changes negligibly during the reaction, we can treat it as constant. The rate law then appears to depend only on the other reactant — this is called pseudo-first order kinetics.
Since [H2O] remains essentially constant, we absorb it into the rate constant:
Rate=k′[ester],where k′=k[H2O]
This is exactly the form of a first order reaction. So the assertion is correct — hydrolysis of an ester follows first order kinetics (under typical conditions).
Step-by-Step Reasoning
-
Identify the true order of the reaction.
The balanced equation shows one molecule of ester reacts with one molecule of water. The fundamental rate law is second order: Rate=k[ester][H2O].
-
Examine the reaction conditions.
In a typical lab or exam context, ester hydrolysis is done in dilute aqueous solution. Water is the solvent — its initial concentration is ~55.5 M and it barely changes because only a tiny fraction is consumed.
-
Apply the concept of excess reactant. …
-
- CBSE 2025Set X11 markMCQQ.An example for pseudo first-order reaction is,(a) The decomposition of gaseous ammonia on a hot platinum surface(b) Photochemical reaction between hydrogen and chlorine(c) Inversion of cane sugar(d) Hydrogenation of ethene
›Reveal solutionSolution
Inversion (hydrolysis) of cane sugar is the standard example of a pseudo first-order reaction — water is in large excess so its concentration is effectively constant.
Hydrolysis of sucrose (cane sugar) into glucose and fructose:
sucroseC12H22O11+H2OH+glucoseC6H12O6+fructoseC6H12O6 …
- CBSE 2025Set A1 markQ.Fill in the blank: The unit of a first order rate constant is ______.
›Reveal solutionSolution
A first-order rate constant always has the unit of (time)⁻¹, e.g. s⁻¹.
For a reaction of order n, the rate constant k has general units of (mol L−1)1−ntime−1. For a first-order reaction (n=1), the concentration term's exponent becomes zero, so the concentration unit cancels out completely, leaving only:
…
- CBSE 2025Set ANNUAL1 markMCQQ.What is the concentration of the reactant in a first order reaction, when the rate of the reaction is 0.6 Ms^-1 and the rate constant is 0.035 s^-1 ?(a) 26.667 M(b) 17.143 M(c) 26.183 M(d) 17.667 M
›Reveal solutionSolution
For a first order reaction, Rate = k[R], so the reactant concentration is simply Rate divided by the rate constant.
For a first order reaction:
Rate=k[R] …
- CBSE 2025Set ANNUAL1 markMCQQ.The units of first order reaction:(a) s⁻¹(b) s(c) mol L⁻¹(d) L⁻¹s
›Reveal solutionSolution
The rate constant of a first order reaction has units of (time)⁻¹, i.e. s⁻¹.
For a general reaction of order n, rate =k[A]n, so
k=[A]nrate=(molL−1)nmolL−1s−1
For a first order reaction (n=1):
…
- CBSE 2025Set ANNUAL1 markMCQQ.Which one of the following is a pseudo first order reaction?(a) Hydrogenation of ethene(b) Hydrolysis of ethyl acetate in the presence of dilute acid(c) Combination of H2 and Br2(d) Decomposition of NH2 on a platinum surface
›Reveal solutionSolution
Water, present in huge excess as solvent, has an essentially constant concentration during the ester's hydrolysis, so the true second-order rate law collapses to an apparent first-order one.
The acid-catalyzed hydrolysis CH₃COOC₂H₅ + H₂O → CH₃COOH + C₂H₅OH is genuinely second order overall (first order in ester, first order in water), but since water is present in vast excess (it is essentially the solvent), its concentration barely changes during the reaction and gets absorbed into the rate constant, so the reaction experimentally behaves as first order — a pseudo …
- CBSE 2024Set ANNUAL1 markMCQQ.The unit of rate constant for a first order reaction is -(a) L2Sec−1(b) Sec−1(c) MolL−1Sec−1(d) Mol−1LSec−1
›Reveal solutionSolution
A first order rate constant has units of (time)−1.
…
- CBSE 2024Set D1 markMCQQ.Which of the following is not a first order reaction?(a) CH3COOCH3 + H2O --H+--> CH3COOH + CH3OH(b) CH3COOC2H5 + NaOH -> CH3COONa + C2H5OH(c) 2H2O2 -> 2H2O + O2(d) 2N2O5 -> 4NO2 + O2
›Reveal solutionSolution
Saponification of an ester by NaOH is second order; the other three are (pseudo/) first order.
- (a) Acid hydrolysis of an ester (with H+ and large excess water) is a pseudo-FIRST-order reaction.
- (b) Ester + NaOH (saponification) depends on both [ester] and [OH-], so rate = k[ester][OH-]: SECOND order.
- (c) Decomposition of H2O2 (2H2O2 -> 2H2O + O2) is first order. …
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