Hydrogen peroxide decomposes in a first order reaction according to:2 H 2 O 2 (aq) ® H 2 O (l) + O 2 (g)The half-life for the decomposition process is t 1/2 = 360 minutes at 38.0 ℃i. (2 marks)Calculate the value for the rate constant, k 1 for the forward reaction at 38.0 ℃ii. (2 marks)If after 1080 mins, the concentration of H 2 O 2 in a sample is measured to be0.154 mol L-1 , calculate the initial concentration of the H 2 O 2 in this sample.iii. (1 mark)Assuming the decomposition follows Arrhenius behaviour, calculate the Arrheniusconstant for this decomposition process if the activation energy is 42 kJ mol -1 .k1
Question
Hydrogen peroxide decomposes in a first order reaction according to:2 H 2 O 2 (aq) ® H 2 O (l) + O 2 (g)The half-life for the decomposition process is t 1/2 = 360 minutes at 38.0 ℃i. (2 marks)Calculate the value for the rate constant, k 1 for the forward reaction at 38.0 ℃ii. (2 marks)If after 1080 mins, the concentration of H 2 O 2 in a sample is measured to be0.154 mol L-1 , calculate the initial concentration of the H 2 O 2 in this sample.iii. (1 mark)Assuming the decomposition follows Arrhenius behaviour, calculate the Arrheniusconstant for this decomposition process if the activation energy is 42 kJ mol -1 .k1
Solution
i. The rate constant (k) for a first order reaction can be calculated using the formula:
k = 0.693 / t1/2
where t1/2 is the half-life of the reaction.
Given that the half-life (t1/2) is 360 minutes, we can substitute this into the formula to find k:
k = 0.693 / 360 min = 0.00193 min^-1
ii. For a first order reaction, the concentration of the reactant at any time can be calculated using the formula:
ln([A]0 / [A]) = kt
where [A]0 is the initial concentration, [A] is the concentration at time t, and k is the rate constant.
Given that [A] = 0.154 mol L^-1, t = 1080 min, and k = 0.00193 min^-1, we can rearrange the formula to find [A]0:
[A]0 = [A] * e^(kt) = 0.154 mol L^-1 * e^(0.00193 min^-1 * 1080 min) = 0.44 mol L^-1
iii. The Arrhenius equation is:
k = A * e^(-Ea/RT)
where A is the Arrhenius constant, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.
Given that Ea = 42 kJ mol^-1 = 42000 J mol^-1, R = 8.314 J mol^-1 K^-1, T = 38.0 ℃ = 311.15 K, and k = 0.00193 min^-1 = 0.00193 / 60 s^-1, we can rearrange the formula to find A:
A = k / e^(-Ea/RT) = 0.00193 / 60 s^-1 / e^(-42000 J mol^-1 / (8.314 J mol^-1 K^-1 * 311.15 K)) = 2.18 x 10^13 s^-1
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