Question

A certain batch of fireflies was observed to "flash" at the rate of 20.8 times per minute at 25°C, and at the lower rate of 5.0 times a minute at a temperature of 15°C.

Assume that the flashing is the result of an overall chemical reaction that has a single rate-limiting step with the highest activation energy. Use this data to estimate the activation energy for this slowest step. You can assume that the concentrations of "reactants" in the fireflies do not depend on temperature.

Answer 1

the activation energy for the slowest step is Ea= 101.7 kJ/mol

Explanation:

Since the concentrations do not vary with temperature, the ratio of reaction rates at different temperatures is the ratio of specific reaction rates:

r₂/r₁ = k₂/k₁ , where 1 represents the scenario with T=25°C and 2 the scenario of 15°C

then knowing Arrhenius equation

k₂=k₀*e^([-Ea/(R*T₂)] and k₁=k₀*e^([-Ea/(R*T₁)]

where k₀= collision factor , Ea= Activation energy , R= ideal gas constant=8.316 J/mol*K, T= absolute temperature

therefore

k₂/k₁ = e^([-Ea/(R*T₂)+Ea/(R*T₁)]

ln (k₂/k₁) = Ea/R (1/T₁- 1/T₂)

Ea = R*ln (k₂/k₁) / (1/T₁- 1/T₂)

replacing values ,

T₂= 25°C= 298K , T₁= 15°C=288K

Ea= R*ln (k₂/k₁) / (1/T₁- 1/T₂) = 8.316 J/mol*K* ln(20.8/5) / (1/288 K - 1/298 K ) = 101740.77 J/mol = 101.7 kJ/mol