 ## General Chemistry

Learn the toughest concepts covered in Chemistry with step-by-step video tutorials and practice problems by world-class tutors

15. Chemical Kinetics

# Arrhenius Equation

The Arrhenius Equation illustrates the temperature dependence of the rate constant k.

Arrhenius Equation
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concept

## Arrhenius Equation Concept 1 1m
Play a video:
the arena's equation investigates the temperature dependence of chemical reaction rates. Here we're going to say that the Iranians equation has an Iranian equation formula and that is K equals a times here. E. Is the inverse of the natural log taken to the negative E. A. Over RT Here, K equals our rate constant. A. Here we're going to say equals our frequency factor are also called our pre exponential factor or sometimes called the Iranians factor. So these are just a bunch of different names for the same variable E equals or activation energy or act on energy of activation or energy barrier. Again, this also has named as well. R equals R gas constant, which here equals 8.314 jewels over molds times. Kaye! Now hear this are constant. We used 8.314 anytime we're referring to speed, energy or velocity. Okay, so keep that in mind. Our becomes 8.314 with the units of jewels over most times K. Anytime one of these three ideas is being discussed. Okay, so keep this in mind. The Arrhenius equation is K equals A times E. To the negative E. A over R. T.
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example

## Arrhenius Equation Example 1 2m
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the gas phase reaction of nitrogen monoxide with chlorine to form en el cl and cl has an activation energy of 7.2 kg joules per mole. And a frequency factor of 8.9 times 10 to the nine molar, it is inverse times seconds, inverse. Here we need to calculate the rate constant at 110°C. Alright, so here we have to figure out what our rate constant is. So we're looking for. Kay and remember what the Iranians equation K. Equals a times E. To the negative E A over R. T. We're told what our frequency factor is. So this is A and that is 8.9 times 10 to the nine polarities, inverse times seconds, inverse Times E to the negative. Now here are uses jewels. So activation energy also needs to be in jewels. So we're going to say we have 7.2 killer jewels per mole. And remember that one killer jewel is equal to 10 to the three jewels. So that's 7200 joules per mole. So that's what we're going to put here. 7200 jewels per mole. Next we're gonna say our is our gas constant 8.314 jewels over moles times K. And then our temperature needs to be in kelvin. So add to 73.15 to our degrees Celsius. And that gives us our kelvin which comes out as 3 83.15 K. So we plugged that in 383.15 Kelvin. If we look here, we're going to say equals so jules cancel out with jules, moles, cancel with moles. Kelvin canceled with kelvin's. So there will be no units on this portion. So the units for our rate constant here will be in malaria, these inverse times seconds inverse when we plug all this in. So we're gonna do here are frequency factor times e raised to this power. When we do that, we're gonna get 9.285 times 10 to the eight polarities, inverse times seconds inverse. Now, within the question, this has to six figs. This has to six figs. This has to six figs. So we need to six figs as our final number of significant figures. So we have 9.3 times 10 to the eight polarities inverse times seconds inverse. So this will be our final answer.
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concept

## Arrhenius Equation Concept 2 2m
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Now the two point form of the Iranians equation shows how changing the temperature can impact the rate constant which uses the variable que. Now here we're going to say that the higher the reaction temperature this causes an increase in the rate constant K. Now here when do we use a two point form of the arena's equation? Well it's used whenever we're dealing with two rate constants and two temperatures for a given reaction. Now the Arrhenius equation 2.4 um formula is as follows, it is Ln K two over K one equals negative E. A. Over R times one over T two minus one over T one. Here. K one is our initial rate constant. K two is our final. T. One is our initial and T two is our final. Now the thing about this is this is the version I want you to remember. This is the one you need to memorize. Professor sometimes can be a little bit tricky. They'll give you two forms of the same exact equation. It doesn't matter which one of you use because mathematically you'll get the same answer. Now you might see it written as Ln K two over K. One equals E. A. Over R. Times one over T one minus one over T two. The equations are pretty similar. Where's the difference? The difference is here we're using negative E. A. And here it is a positive E. A. Here we're using T. Two. First. Here we're using T. One first. Again it doesn't matter which one of the two that you use both will give you the same exact answer. Whether you're solving for one of the K. Is one of the T. S. Or activation energy itself. I just prefer to use this top one here because this is the one that is shown most often. Okay, so just keep that in mind. You might see it in two different ways in class on a formula sheet. What? Whatever, But use the top one. Alright, so keep in mind. This is the two point form of the Arrhenius equation. We use it anytime. We're dealing with two rate constant K. S. Or two temperatures T.

When dealing with TWO rate constants or TWO temperatures then we use the Two-Point Form of the Arrhenius Equation.

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example

## Arrhenius Equation Example 2 3m
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a chemical reaction has rate constants of 4.6 times 10 to the negative two seconds inverse and 8.1 times 10 to negative two seconds inverse at zero degrees Celsius and 20 degrees Celsius respectively. What is the value of the activation energy? Alright, so they're giving us to rate constants since this one has stated first, this is K one And this would have to be K2, respectively. Would mean that zero degrees is connected to K one and therefore would be T one And the second temperature will be connected to K two and therefore it is T two. Now we'd say here that Ln of K two over K one equals negative E A over R times one over T two minus one over T one. Here, we're gonna place the numbers that we have. So it's Ln of so K two is 8.1 times 10 to the negative two, divided by 4.6 times 10 to the negative two. We're looking for activation energy. So, e is what we need to find our here is our gas constant, which is 8.314 jewels over moles, times K times We're gonna say one over T two. So remember we have to add to 73.15 to each one of these Calvin temperatures. So when we do that, this becomes to 93.15 k minus one over To 73.15 K. Mhm. Here we're going to do Ln of eight K two over K one. When we do that we're going to get as our answer 10.4 point 565808 equals. Alright. So E. A. It's gonna be multiplying by what's in here. So when we do that we're gonna get uh negative value in here. So this is gonna be what's in here is equal to negative 2.49769 times 10 to the negative four A negative times a negative gives me positive. That'd be positive E. A. Times 2.49769 times 10 to the negative four Divided by still 8.314. Here we're going to multiply both sides by 8.314. So when we do that we're gonna get 4. equals E. A. Still times 2.49769 times 10 to the negative four, divide both sides now By 2.4976. 9 times 10 to the -4. So when we do that we're gonna get our E. A. Which I'm gonna right over here. So er here equals 18,833.91 jules per mole. Now if we go back up here this has two sig figs to 6 to figs for 4.6 and 8.1. Let's not worry about the temperatures. So let's just do this in terms of two sig figs. So that will become 1.9 times 10 to the four jewels per mole. So this would be our value for activation energy, abbreviated as e a.
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concept

## Arrhenius Equation Concept 3 1m
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So we use the linear form of the Arrhenius equation when a plot of lnk versus inverse temperature is given. And we're gonna say it's used to determine the activation energy or E. A. Of the reaction. Now recall that the equation for straight line is Y equals mx plus B. And what we're gonna do is we're going to rearrange the Iranians equation to fit this equation for a straight line. Now if we take a look at a plot, remember our X. Here is represented by inverse temperature. Ry by Ellen K. Alan represents our starting point. And remember our slope is changing y over change in X which is rise over run by basically rearranging our Iranians equation. Were able to get the linear form of it here. This linear form is related to a straight line, it's equal to Y equals mx plus B. Here are Y is alan Kay. Our slope M is related to negative ea over R. One over T R. Inverse temperature is equal to X and L N A. Is B. Which is again our starting point. Now here we say, remember these variables that K. Is our rate constant A equals our frequency factor or pre exponential factor. E. A. Is our activation energy and our is our gas constant which is equal to 8.314 jewels over moles times K. So remember we use this linear form of the Arrhenius equation anytime we're given a plot of lnk versus inverse temperature

In order to relate the plot of a graph to the Arrhenius equation then we manipulate it into the Linear Form of the equation.

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example

## Arrhenius Equation Example 3 1m
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A plot of LNK vs Inverse temperature has a slope of negative 8313. What is the activation energy for this reaction? So remember when they give us a plot, it's always of Y versus X. And the fact that we're dealing with lnk versus inverse temperature means that we're going to deal with the linear form of the Iranians equation. So here we're going to say that Ellen K equals negative E A over R times one over T plus Ln of A. And remember this is equal to the formula for straight line, which is why equals mx plus B. M represents our slope. And it's also equal to negative activation energy divided by R. R gas constant. So them telling us our slope here is them really giving us em we're gonna set them equal to each other Here. This is -8313 Calvin equals negative E A over R R S R. Gas constant 8.314 jules over most times K. We're going to multiply both sides by 8.314 jewels over moles, times K. Kelvin's cancel out and we'll have our answer in joules per mole. Now here, initially we're going to have is negative 69,114.282 jewels per mole V equals negative E A. Just divide both sides by -1 and we'll get e equals here. This has four sig figs in its or answers should have four sig figs as well. So we're gonna say 69,110 jewels per mole as our final answer.
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Problem

The rate constant of a reaction at 32°C is 0.060/s. If the frequency factor is 3.1 × 1015 s–1, what is the activation barrier?

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Problem

A reaction with an activation energy Ea = 55.00 kJ/mol is run at temperature of 30ºC. Determine the temperature required to increase the rate constant 3 times.

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Problem

The following data shows the rate constant of a reaction measured at numerous temperatures. Use the Arrhenius plot to determine the frequency factor for the reaction.  