From the following rate constants, determined at five temperat...

Question

From the following rate constants, determined at five temperatures, calculate the experimental energy of activation and ∆G^‡, ∆H^‡, and ∆S^‡ for the reaction at 30 °C:

Accepted Answer

All right. Hello everyone. So this question says that the rate constants of a certain reaction at different temperatures were recorded as Chamblee calculate the activation energy delta G double dagger, delta H double dagger and delta S double dagger of the reaction at 50 °C recall that the double dagger notation indicates the transition state. All right. So in this case, based on the data table provided 50 °C is not actually given to us as a data point. So because of this, we're going to have to calculate it using the linear form of the erroneous equation. We can use this to go ahead and calculate the activation energy to start off. So recall first and foremost that the linear form of the erroneous equation is as follows. We're the Ln of Kay is equal to negative E sub A divided by R multiplied by one over T and then added to the Ln of a. Notice how this equation follows the equation or follows the form of Y equals MX added to be. So why is the Ellen of K M which is the slope is equal to negative E sub A divided by R X is equal to one over T and the Ln of A is the Y intercept. So B is equal to the Ln of A. Now using this information, we can go ahead and make a graph of X versus Y. In other words, the inverse of the temperature versus the Ln of K. Keep in mind, however, that for this graph, the temperature must be in units of Kelvin. But after plotting this graph, we can go ahead and see the equation of the linear form. And when we do so the equation given is that Y is equal to negative 10,764 X added to 25.746. So in this case, M is equal to negative 10,764. So negative 10,764 is equal to negative E sub A divided by R recall that E sub A refers to the activation energy and R is the universal gas constant. So here negative 10,764 is equal to negative E sub A divided by art gas constant which is 1.96 multiplied by 10 to the negative third power kilo calories per mole Kelvin. So at this point, we would go ahead and evaluate the activation energy by multiplying the value of the slope by the gas constant and then multiplying by negative one. This means that E of A is equal to 21.3 kilo calories per m. So here is one of our answers. And at this point, we can go ahead and continue by finding or calculating delta G double dagger. Now recall that in the equation to evaluate delta G double dagger, the rate constant is required at the specified temperature. So before we can go ahead and use that equation, let's go ahead and calculate the rate constant at 50 °C using the equation obtained from the graph. So here, because temperature must be in Kelvin or T is equal to 50.0 °C added to 273.15. This equals 323.15 Kelvin. So because the inverse of the temperature is our X value, we can go ahead and plug this into our equation using the activation energy to calculate Y which is the Ln of K. So here Y is equal to negative 10,764 multiplied by one divided by 323.15 added to 25.746. This results in A Y value which is equal to the Ln of K equal to negative 7.5 636. So to go ahead and evaluate K which is the rate constant at 50 °C K is equal to E to the power of negative 7.5636. And so the rate constant, ultimately equals 5.19 multiplied by 10 of the negative fourth power in verse seconds. So now that we have the rate constant at the appropriate temperature, we can go ahead and proceed with the equation to find delta G double dagger. Recall the, the negative delta G double dagger is equal to RT multiplied by the Ln of K divided by T K sub B. So here there are a few different constants that we have to recall. Here R is once again, the gas constant K is the rate constant that was calculated in the prior step. However, H is planks constant which is equal to 1.5 eight multiplied by 10 to the negative 31st power kilo calorie seconds. On the other hand, K sub B is Boltzmann's constant, which is equal to 3.30 multiplied by 10 to the negative 19th power kilo calories per Kelvin. So using all of our given quantities, we can go ahead and solve for negative delta G double dagger. So negative delta G double dagger is equal to R. That's 1.96 multiplied by 10 to the negative third power kilo calories per mole. Kelvin multiplied by T that's 323.15 Kelvin and multiplied by the natural logarithm of K which was 5.19 multiplied by 10 to the negative fourth in verse seconds multiplied by H which is 1.58 multiplied by 10 to the negative 31st kilo calorie seconds divided by the temperature. Once again, that's 323.15 Kelvin and multiplied by Boltzmann constant, which is 3.30 multiplied by 10 to the negative 19th kilo calories per Kelvin. All right. So when valuing or when evaluating, excuse me, this entire expression and subsequently multiplying that quantity by negative one delta G double dagger, that's delta G of the transition state is ultimately equal to 26.7 kilo calories per mole. And so from here, we can continue on to calculate delta H double dagger and delta S double dagger. So recall the delta H double dagger is equal to E sub a subtracted by RT. So in this case, that's euba the activation energy which was 21.3 kilo calories per more subtracted by the gas constant R which was 1.96 multiplied by 10 to the negative third kilo calories per mole. Kelvin multiplied by T which was 323.15. Kelvin. I realize that its cutting off of the screen. So I am going to go ahead and make this smaller. So after evaluating this expression, delta H double dagger is equal to 20.7 kg calories per month. Now, I do want to point out here that all of the answers have been rounded to three significant figures to match the number of significant figures provided in each of the data values in the beginning of this video. But with that being said, there's only one more value to calculate now. So lets talk about delta S double dagger. So delta S double dagger is equal to delta H double dagger subtracted by delta G double dagger divided by T. So that would be 20.7 kilo calories per more subtracted by 26.7 kilo calories per more divided by our temperature of 323 0.15. Kelvin. This means that delta S double dagger is ultimately equal to negative 0.0186 kilo calories per more. Kelvin. And there you have it once again, my writing was starting to cut off from the screen. So I wanted to make that a little bit smaller before highlighting, but here we have calculated all four values. So if you watch this video all the way through, thank you so very much. And I hope you found this helpful.

From the following rate constants, determined at five temperatures, calculate the experimental energy of activation and ∆G^‡, ∆H^‡, and ∆S^‡ for the reaction at 30 °C:
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Key Concepts

Arrhenius Equation

Gibbs Free Energy of Activation (∆G‡)

Enthalpy and Entropy of Activation (∆H‡ and ∆S‡)

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