Arrhenius equation and rate of change

[Big-Daddy]

I wanted to see how the rate of change of rate constant with temperature (dk/dT) changes with activation energy. I tried to do this with differentials: k=A*e^-E_A/RT so

$$\frac{dk}{dT} = \frac{A E_A \cdot e^{-\frac{E_A}{RT}} } {RT^2}$$

and then

$$\frac{d(\frac{dk}{dT})}{dE_A} = A \cdot e^{-\frac{E_A}{RT}} \cdot (\frac{1}{RT^2} - \frac{1}{R^2T^3})$$

So long as E_A>RT, it appears that rate of change of rate constant with temperature would decrease for increasing activation energy (i.e. if you increase activation energy, then the rate constant is less susceptible to increasing when you change the temperature). E_A>RT would be almost universally the case I imagine - at 298 K that's less than 2.5 kJ/mol.

But this doesn't tally up with what we know, which is that for a given temperature increase, a greater activation energy leads to a greater increase in rate constant. e.g. if we write well-known

k₂/k₁ = exp( -E_A/R * (1/T₂² - 1/T₁²) )

and try some values we get this conclusion. So what went wrong or what is wrong with my idea to do it by differentiation?

 

[Big-Daddy]

k₂/k₁ = exp( -E_A/R * (1/T₂² - 1/T₁²) )

Oops I made a stupid mistake here. Sorry if this was confusing people. It should be

k₂/k₁ = exp( E_A/R * (1/T₁ - 1/T₂) )

No squared terms.

 

[Chestermiller]

I wanted to see how the rate of change of rate constant with temperature (dk/dT) changes with activation energy. I tried to do this with differentials: k=A*e^-E_A/RT so

$$\frac{dk}{dT} = \frac{A E_A \cdot e^{-\frac{E_A}{RT}} } {RT^2}$$

and then

$$\frac{d(\frac{dk}{dT})}{dE_A} = A \cdot e^{-\frac{E_A}{RT}} \cdot (\frac{1}{RT^2} - \frac{1}{R^2T^3})$$

So long as E_A>RT, it appears that rate of change of rate constant with temperature would decrease for increasing activation energy (i.e. if you increase activation energy, then the rate constant is less susceptible to increasing when you change the temperature). E_A>RT would be almost universally the case I imagine - at 298 K that's less than 2.5 kJ/mol.

But this doesn't tally up with what we know, which is that for a given temperature increase, a greater activation energy leads to a greater increase in rate constant. e.g. if we write well-known

k₂/k₁ = exp( -E_A/R * (1/T₂² - 1/T₁²) )

and try some values we get this conclusion. So what went wrong or what is wrong with my idea to do it by differentiation?

You differentiated with respect to E_A incorrectly. Try again. The two terms in parenthesis do not have consistent units.

 

[Big-Daddy]

You differentiated with respect to E_A incorrectly. Try again. The two terms in parenthesis do not have consistent units.

Thanks, I had indeed made this mistake - the actual differential should be

$$\frac{d(\frac{dk}{dT})}{d(E_A)} = \frac{A}{RT^2} \cdot e^{-\frac{E_A}{RT}} \cdot (1 - \frac{E_A}{RT})$$

But it seems that my original conclusion still holds true - that E_A < RT (incredibly small activation energy) is necessary for rate of change of rate constant with temperature to increase at a higher activation energy, which however is the trend which the expression for k₂/k₁ indicates is true for all E_A (it seems to me)?