A popular chemical demonstration is the "magic genie" procedure, in which hydrogen peroxide decomposes to water and oxygen gas with the aid of a catalyst. The activation energy of this (uncatalyzed) reaction is 70.0 \(\mathrm{kJ} / \mathrm{mol}\) . When the catalyst is added, the activation energy (at \(20 .^{\circ} \mathrm{C} )\) is 42.0 \(\mathrm{kJ} / \mathrm{mol}\) . Theoretically, to what temperature ( \((\mathrm{C})\) would one have to heat the hydrogen peroxide solution so that the rate of the uncatalyzed reaction is equal to the rate of the catalyzed reaction at \(20 .^{\circ} \mathrm{C} ?\) Assume the frequency factor \(A\) is constant, and assume the initial concentrations are the same.

Step by step solution

01

Write down the Arrhenius equation

The Arrhenius equation relates the rate constant (k) of a chemical reaction to its activation energy (Ea), temperature (T), and the pre-exponential factor (A) as follows: \[k = Ae^{\frac{-Ea}{RT}}\] where R is the gas constant (8.314 J/mol·K)

02

Set up the equation to relate the rates of uncatalyzed and catalyzed reactions

We're given the activation energies of the uncatalyzed (Ea1) and catalyzed reactions (Ea2), and we want the rates to be equal. Therefore, we'll equate the Arrhenius equations for each reaction, assuming the frequency factors and initial concentrations are the same: \[k_1 = k_2\] \[Ae^{\frac{-Ea1}{R(T+273)}} = Ae^{\frac{-Ea2}{R(20+273)}}\]

03

Solve for temperature

By equating the expressions in step 2, we can cancel out the pre-exponential factor (A) and solve for temperature (T) using the known values of the activation energies (Ea1 = 70.0 kJ/mol and Ea2 = 42.0 kJ/mol) and the gas constant R: \[e^{\frac{-Ea1}{R(T+273)}} = e^{\frac{-Ea2}{R(20+273)}}\] To solve for T, we'll first take the natural logarithm of both sides: \[\frac{-Ea1}{R(T+273)} = \frac{-Ea2}{R(20+273)}\] Next, we'll multiply both sides by -R and simplify: \[(T + 273)(Ea1) = (20 + 273)(Ea2)\] Now, solve for T: \[T = \frac{(20+273)(Ea2)}{Ea1} - 273\] Plug in the given values of Ea1 and Ea2: \[T = \frac{(20+273)(42.0 \mathrm{kJ/mol})}{70.0 \mathrm{kJ/mol}} - 273\]

04

Calculate the temperature

Calculate T using the values obtained in step 3: \[T = \frac{(293)(42.0 \mathrm{kJ/mol})}{70.0 \mathrm{kJ/mol}} - 273\] \[T \approx 92.6^{\circ} \mathrm{C}\] So, the temperature to which one would have to heat the hydrogen peroxide solution so that the rate of the uncatalyzed reaction is equal to the rate of the catalyzed reaction at 20°C is approximately 92.6°C.

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