Question

Read the following passage and answer the questions that follow:

The rate of reaction is concerned with decrease in the concentration of reactants or increase in the concentration of products per unit of time. It can be expressed as instantaneous rate at a particular instant of time and average rate over a large interval of time. A number of factors such as temperature, concentration of reactants, catalyst affect the rate of reaction. Mathematical representation of rate of a reaction is given by rate law: Rate = k[A]x [B]y x and y indicate how sensitive the rate is to change in concentration of A and B. Sum of x + y gives the overall order of a reaction. When a sequence of elementary reactions gives us the products, the reaction is called complex reaction. Molecularity and order of an elementary reaction are same. Zero-order reactions are relatively uncommon but they occur under special conditions. All natural and artificial radioactive decay of unstable nuclei takes place by first-order kinetics.

1. What is the effect of temperature on the rate constant of a reason?    [1]
2. For a reaction \[\ce{A + B → Product}\], the rate law is given by, Rate = k[A]² [B]^1/2. What is the order of the reaction?    [1]
3. How order and molecularity are different for complex reactions?    [1]
4. A first-order reaction has a rate constant 2 × 10^–3 s^–1. How long will 6 g of this reactant take to reduce to 2 g?    [2]
   OR
   The half-life for radioactive decay of ¹⁴C is 6930 years. An archaeological artifact containing wood had only 75% of the ¹⁴C found in a living tree. Find the age of the sample.
   [log 4 = 0.6021, log 3 = 0.4771, log 2 = 0.3010, log 10 = 1]    [2]

Answer 1

a. The rate constant (k) for a reaction increases with an increase in temperature and becomes almost double with every 10° rise in temperature. This effect is expressed by the Arrhenius equation:

k = `Ae^(-E_a//RT)`

b. According to the equation, r = k[A]²[B]^1/2

Order of the reaction = `2 + 1/2 = 5/2`

c. Order of reaction is defined as the sum of the powers of the molar concentration of the reaction species in the rate equation of the reaction. It is applicable to both elementary and complex reactions.

Molecularity of a reaction is defined as the total number of reacting species participating in an elementary reaction. It has no significance for complex reactions as applicable only to elementary reactions.

d. k = 2 × 10^–3 s^–1 

t = `2.303/k log  [R]_0/[R]`

t = `2.303/(2 xx 10^-3)  log  6/2`

t = 1151.5 × 0.4771

t = 549.3 s

OR

According to first-order reaction,

Half-life (t_1/2) = `0.693/k`

t_1/2 = 6980 years

k = `0.693/6980`

However, the time taken can be calculated using the first-order rate of reaction when wood contains only 75% of ¹⁴C.

Initial concentration of C₁₄, [R]₀ = 100

Amount at time t s, [R] = 75

t = `2.303/k log  [R]_0/([R])`

t = `2.303/(0.693/6980) log  100/75`

t = 2898 years

Thus, the age of the sample is approximately 2898 years.