Arrhenius equation and rate of change

In summary, the conversation discussed the rate of change of rate constant with temperature and activation energy. The speaker attempted to use differentials to determine this relationship but made a mistake in their calculations. The correct differential shows that for a given temperature increase, a greater activation energy leads to a greater increase in rate constant. However, the original conclusion that EA < RT is necessary for this relationship to hold still stands.
  • #1
Big-Daddy
343
1
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-EA/RT so

[tex]\frac{dk}{dT} = \frac{A E_A \cdot e^{-\frac{E_A}{RT}} } {RT^2}[/tex]

and then

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

So long as EA>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). EA>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( -EA/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?
 
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  • #2
Big-Daddy said:
k2/k1 = exp( -EA/R * (1/T22 - 1/T12) )

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

k2/k1 = exp( EA/R * (1/T1 - 1/T2) )

No squared terms.
 
  • #3
Big-Daddy said:
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-EA/RT so

[tex]\frac{dk}{dT} = \frac{A E_A \cdot e^{-\frac{E_A}{RT}} } {RT^2}[/tex]

and then

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

So long as EA>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). EA>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( -EA/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 EA incorrectly. Try again. The two terms in parenthesis do not have consistent units.
 
  • #4
Chestermiller said:
You differentiated with respect to EA 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

[tex]\frac{d(\frac{dk}{dT})}{d(E_A)} = \frac{A}{RT^2} \cdot e^{-\frac{E_A}{RT}} \cdot (1 - \frac{E_A}{RT})[/tex]

But it seems that my original conclusion still holds true - that EA < 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 EA (it seems to me)?
 
  • #5


Your approach to using differentials to analyze the Arrhenius equation is a valid one, but there are a few things to consider.

Firstly, the Arrhenius equation is an empirical equation and its validity is limited to a specific range of temperatures. It is not a fundamental law and may not hold true for all reactions and temperature ranges. Therefore, the results you obtained from your differentiation may not accurately reflect the behavior of the rate constant with temperature.

Secondly, your approach assumes that the activation energy (EA) is constant as the temperature changes, which is not always the case. In reality, the activation energy may also change with temperature, which can affect the rate of change of the rate constant with temperature.

Lastly, your approach only considers the effect of temperature on the rate of change of the rate constant, but there are other factors that can affect this relationship, such as the nature of the reactants, the presence of catalysts, and the overall reaction mechanism.

In conclusion, while your approach is a valid attempt to analyze the Arrhenius equation, it is limited in its applicability and may not accurately reflect the behavior of the rate constant with temperature. It is important to consider all factors and limitations when analyzing scientific equations and their relationships.
 

1. What is the Arrhenius equation?

The Arrhenius equation is a mathematical equation that describes the relationship between the rate of a chemical reaction and the temperature of the system. It is named after Swedish chemist Svante Arrhenius, who first proposed it in 1889.

2. How does the Arrhenius equation relate to the rate of change?

The Arrhenius equation states that the rate of a chemical reaction increases exponentially with an increase in temperature. This means that as the temperature increases, the rate of change also increases, resulting in a faster reaction.

3. What are the key components of the Arrhenius equation?

The Arrhenius equation includes three main components: the pre-exponential factor (A), the activation energy (Ea), and the gas constant (R). These components represent the frequency of collisions between reactant molecules, the energy required for the reaction to occur, and the temperature of the system, respectively.

4. How is the Arrhenius equation used in chemistry?

The Arrhenius equation is used to calculate the rate constant (k) of a reaction, which is a measure of how fast the reaction takes place. It is also used to predict the effect of temperature on the rate of a reaction and to determine the activation energy of a reaction.

5. What are the limitations of the Arrhenius equation?

The Arrhenius equation assumes that the reaction is a simple two-step process and that the only factor affecting the rate of the reaction is temperature. In reality, there can be other factors such as catalysts or changes in concentration that can also affect the rate of the reaction. Additionally, the Arrhenius equation only applies to reactions that follow first-order kinetics.

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