Answers:
(a) Second order in A
(b) Second order in B
(c) r = k[A]²[B]²
(d) k = 0.0123 L³mol⁻³s⁻¹
(e) [A] = 8.86 mol·L⁻¹
Step-by-step explanation:
A + B ⟶ C
The rate law is r = k [A]^m[b]ⁿ
Our problem is to determine the values of m and n.
We use the method of initial rates to determine the order of reaction with respect to a component.
(a) Order with respect to A
We must find a pair of experiments in which [A] changes, but [B] doesn't.
They would be Experiments 3 and 1.
[B] is constant, so only [A] is changing the rate
r₃/r₁ = {k[A]₃^m}/{k[A]₁^m } Cancel the ks
r₃/r₁ = {[A]₃/[A]₁}^m
80/5 = (40.2/10.1)^m
16 = 3.98^m
(i) Method 1: by inspection
16 ≈4²
If quadrupling the concentration multiplies the rate by a factor of 16 (4²), the reaction is 2nd order.
(ii) Method 2: mathematical
16 = 3.98^m
log16 = mlog3.98 Take the logarithm of each side
m = (log16)/(log3.98)
= 1.204/0.5999
= 2.007 Round off to nearest integer
≈ 2
By either method, the reaction is second order in A.
r = k[A]²
(b) Order with respect to B
We must find a pair of experiments in which [B] changes, but [A] doesn't. There are none.
However, we know the effect of A on the rate.
Choose a different pair, say, Experiments 2 and 1.
r₂/r₁ = ([A₂]/[A]₁)²([B]₂/[B₁]^m
80/5 = (19.8/10.1)²(3.99/2.01)^m
16 = (1.960)²(1.985)ⁿ
16 = 3.843 × 1.985ⁿ Divide each side by 3.843
4.163 = 1.85ⁿ Take the logarithm of each side
log4.163 = nlog1.985 Divide each side by log1.985
n = log4.163/log1.985
= 2.08
≈ 2
The reaction is second order in B.
rate = k[B]²
(c) Overall rate law
The reaction is second order in A and second order in B.
The overall rate law for the reaction is
r = k[A]²[B]²
(d) Value of k
Choose any experiment (say, Experiment 3) and insert the known values.
r = k[A]²[B]²
80 mol·L⁻¹·s⁻¹ = k(40.2 mol·L⁻¹)²(2.00 mol·L⁻¹)²
= k × 1616 mol²L⁻² × 4.00 mol²L⁻²
= k × 6464 mol⁴L⁻⁴ Divide each side by 6464 mol⁴L⁻⁴
k = (80 mol·L⁻¹s⁻¹)/(6464 mol⁴L⁻⁴)
= 0.0123 L³mol⁻³s⁻¹
(e) [A] in Experiment 4
r = k[A]²[B]²
35 = 0.0123 × [A]²[6.00]²
= 0.4456[A]² Divide each side by 0.4456
[A]² = 35/0.4456
= 78.6 Take the square root of each side
[A] = 8.86 mol·L⁻¹
