CH 203 Problem 14.56

14.56 Understanding the high-temperature behavior of nitrogen oxides is essential for controlling pollution generated from automobile engines. The decompsition of nitric oxide (NO) to N₂ and O₂ is second-order with a rate constant of 0.0796 M^-1s^-1 at 737^oC and 0.0815 M^-1s^-1 at 947oC. Calculate the activation energy for the reaction.

Alrighty. Please fasten your seat belt and return your tray to the upright position:) Just kidding. Let's figure out what to do here......

The Arrhenius equation relates the rate constant k to temperature as follows:

k = Ae^-Ea/RT

Take a look: if we know Ea and A, then we can calculate k at some temperature T, no? Here, we know k at a couple of temperatures, so we're not just ready to plug and chug yet. If we knew A, we could substitute k, T, and A and find Ea, but we don't know A. So what to do?

Start by taking the natural log of both sides of the Arrhenius equation: we get

now, let's say we know the rate constant k₁ at some temperature T₁. We write down

and let's say we know the rate constant k at some other temperature T₂ (so we have k₂ and T₂):

now, let's take our equation for ln k₁ and subtract the equation for ln k₂ from it.

we assume that the collision factor A is the same at both temperatures; it goes away, and we can clean this up a little bit: (be sure you follow the algebra here)

now, we know k₁, k₂, T₁, and T₂, and we can find Ea:

so

DAC 8/6/08