Solved Numericals on Zero, First, Second and Third Order reactions | Physical Chemistry PDF Download

Q.1. Derive an expression for the rate constant for a first order reaction.

Rate constant of a first order reaction: The reaction in which, the overall rate of the reaction is proportional to the first power of concentration of one of the reactants only are called first order reaction.

where k₁ is the rate constant of the first order reaction.

At the beginning of the reaction, time f = 0, let the concentration of A be 'a' mole lit^-1. After the reaction has proceeded for some time't', let the concentration of A that has reacted be x mole lit-1. The concentration of unreacted A remaining at time't' will be (a - x) mole lit^-1. The rate of the reaction will be dx/dt. For a first order reaction,
rate = dx/dt = k₁ (a - x ) ..... (i)
Integrating (1), both sides

Which is -In (a - x) = k₁t + c .....(2)
C = integration constant,
at time, t = 0, x = 0
In equation (2)
-ln (a -0 ) = k₁ x 0 + c
or
c = ln a
Substituting c value in equation (2)
-ln (a -x ) = k₁t- In a
rearranging,
 
Unit of k₁ is sec ^-1
This equation is known as the first order rate constant equation.

Q.2. The half-life for radioactive decay of ¹⁴C is 5730 years. An archaeological artifact containing wood had only 80% of the ¹⁴C found in a living tree. Estimate the age of the sample.

Decay constant k=

The rate of counts is proportional to the number of C-14 atoms in the sample.

N₀ =100, N=80
The age of the sample t =

Q.3. The unit of rate constant for zero order reaction is:
(a) s⁻¹
(b) mol L⁻¹ s⁻¹
(c) L mol⁻¹ s⁻¹
(d) L² mol⁻²s⁻¹

Q.4. The rate for the decomposition of NH₃ on platinum surface is zero order. What are the rate of production of N₂ and H₂ if K= 2.5 x 10^-4 mol litre^-1s^-1.

2NH₃ →N₂ +3H₂
Rate of reaction =
For a zero order reaction
Rate = k = 2.5 x 10^-4 M/s
Rate of production of N₂ =
Rate of production of H₂ =

Q.5. Show that in a first order reaction, time required for completion of 99.9% is 10 times of half-life (t_1/2) of the reaction.

For first order reaction,

Given that:
x = 99.9 a for t_99.9%
x = 0.5 a for t_50%




⇒ t_99.9% = 10 × t_50%
Hence proved.

Q.6. The rate constant of the reaction A→B is 0.6×10⁻³  moleL⁻¹ s⁻¹ . If the concentration of A is 5 M, then concentration of B after 20 minutes is:
(a) 0.36 M
(b) 0.72M
(c) 1.08M
(d) 3.60M

Correct option is Option (b)

The rate constant has unit mole per second which indicates zero order reaction.

x = kt

The concentration of B after 20 minutes is 20 x 60 x 0.6 x 10^-3 = 0.72M

Q.7. For a first order reaction, show that time required for 99% completion is twice the time required for the completion of 90% of reaction.

For a first order reaction, we have

Let a M be the initial concentration.
When reaction is 99% complete,



When reaction is 90% complete,

Divide equation (1) by (2), we get

Q.8. A first order reaction takes 40 min for 30% decomposition. Calculate t_1/2.

Let a M be the initial concentration. After 40 minutes, the concentration is a

The half life period,

Q.9. The rate constant for a first order reaction is 60 s⁻¹ . How much time will it take to reduce the initial concentration of the reactant to its 1/16^th value?

Given that the rate constant for a first order reaction is 60 s ⁻¹ .
Let a M be the initial concentration.
Final concentration will be
Rate constant, k=60/s


t=4.6 × 10 ⁻²  seconds

Q.10. A first order reaction has a rate constant 1.15×10 ⁻³ s ⁻¹ . How long will 5 g of this reactant take to reduce to 3 g?
(a) 444 s
(b) 400 s
(c) 528 s
(d) 669 s

Correct Answer is option (a)

= 2.00 × 10³ log(1.667)

= 2 × 10³ × 0.2219 = 444 s.