Consider a hypothetical reaction between a, b, and c that is first order in a, zero order in b, and second order in c. (a) write the rate law for the reaction. (b) how does the rate change when [a] is doubled and the other reactant concentrations are held constant? (c) how does the rate change when [b] is tripled and the other reactant concentrations are held constant? (d) how does the rate change when [c] is tripled and the other reactant concentrations are held constant? (e) by what factor does the rate change when the concentrations of all three reactants are tripled? (f) by what factor does the rate change when the concentrations of all three reactants are cut in half?

(a) Rate = k[A][B]⁰[C]² = k[A][C]².

(b) The rate of the reaction will be doubled as the the rate of the reaction will be doubled.

(c) The rate of the reaction will not be changed as the rate of the reaction is zero order for B and does not depend on the concentration of reactant B.

(d) The rate of the reaction will increase by a factor of 9 when [C] is tripled and the other reactant concentrations are held constant.

(e) The rate of the reaction will increase by a factor of 27 the concentrations of all three reactants are tripled.

(f) The rate of the reaction will decrease by a factor of 8  the concentrations of all three reactants are cut in half.

Explanation:

(a) Write the rate law for the reaction.

The rate of the reaction is the change in the concentration of the reactants (decrease) or the products (increase) with time.

Rate = ΔConcentration/Δt.

The rate of the reaction is directly proportional to the concentration of the reactants.

R ∝ [A][B][C].

Since the mentioned reaction is first order in A, zero order in B, and second order in C.

∴ Rate = k[A][B]⁰[C]² = k[A][C]²,

where, k is the rate constant of the reaction.

(b) How does the rate change when [A] is doubled and the other reactant concentrations are held constant?

If we consider the concentration of [A₁] = [A₀].

As the concentration of [A] is doubled, [A₂] = [2A₀]. [B] and [C] are held constant.

∴ Rate₁ = k[A₁][B₁]⁰[C₁]² = k[A₀][B₀]⁰[C₀]²

Rate₂ = k[A₂][B₂]⁰[C₂]² = k[2A₀][B₀]⁰[C₀]²

∴ Rate₁/Rate₂ = (k[A₀][B₀]⁰[C₀]²)/(k[2A₀][B₀]⁰[C₀]²) = 1/2.

∴ Rate₂ = 2(Rate₁).

So, the rate of the reaction will be doubled.

(c) How does the rate change when [B] is tripled and the other reactant concentrations are held constant?

If we consider the concentration of [B₁] = [B₀].

As the concentration of [A] is doubled, [B₂] = [3B₀]. [A] and [C] are held constant.

∴ Rate₁ = k[A₁][B₁]⁰[C₁]² = k[A₀][B₀]⁰[C₀]²

Rate₂ = k[A₂][B₂]⁰[C₂]² = k[A₀][3B₀]⁰[C₀]²

∴ Rate₁/Rate₂ = (k[A₀][B₀]⁰[C₀]²)/(k[A₀][3B₀]⁰[C₀]²) = 1.

∴ Rate₂ = (Rate₁).

So, the rate of the reaction will not be changed as the rate of the reaction is zero order for B and does not depend on the concentration of reactant B.

(d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant?

If we consider the concentration of [C₁] = [C₀].

As the concentration of [A] is doubled, [C₂] = [3C₀]. [A] and [B] are held constant.

∴ Rate₁ = k[A₁][B₁]⁰[C₁]² = k[A₀][B₀]⁰[C₀]²

Rate₂ = k[A₂][B₂]⁰[C₂]² = k[A₀][B₀]⁰[3C₀]²

∴ Rate₁/Rate₂ = (k[A₀][B₀]⁰[C₀]²)/(k[A₀][B₀]⁰[3C₀]²) = 1/9.

∴ Rate₂ = 9(Rate₁).

So, the rate of the reaction will increase by a factor of 9.

(e) By what factor does the rate change when the concentrations of all three reactants are tripled?

If we consider the concentration of [A₁] = [A₀], [B₁] = [B₀], and [C₁] = [C₀].

As the concentration of all three reactants are tripled:

[A₂] = [3A₀], [B₂] = [3B₀], and [C₂] = [3C₀].

∴ Rate₁ = k[A₁][B₁]⁰[C₁]² = k[A₀][B₀]⁰[C₀]²

Rate₂ = k[A₂][B₂]⁰[C₂]² = k[3A₀][3B₀]⁰[3C₀]²

∴ Rate₁/Rate₂ = (k[A₀][B₀]⁰[C₀]²)/(k[3A₀][3B₀]⁰[3C₀]²) = 1/27.

∴ Rate₂ = 27(Rate₁).

So, the rate of the reaction will increase by a factor of 27.

(f) By what factor does the rate change when the concentrations of all three reactants are cut in half?

If we consider the concentration of [A₁] = [A₀], [B₁] = [B₀], and [C₁] = [C₀].

As the concentration of all three reactants are tripled:

[A₂] = [1/2A₀], [B₂] = [1/2B₀], and [C₂] = [1/2C₀].

∴ Rate₁ = k[A₁][B₁]⁰[C₁]² = k[A₀][B₀]⁰[C₀]²

Rate₂ = k[A₂][B₂]⁰[C₂]² = k[1/2A₀][1/2B₀]⁰[1/2C₀]²

∴ Rate₁/Rate₂ = (k[A₀][B₀]⁰[C₀]²)/(k[1/2A₀][1/2B₀]⁰[1/2C₀]²) = 8.

∴ Rate₂ = 1/8(Rate₁).

So, the rate of the reaction will decrease by a factor of 8.

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