Ethyl chloride vapor decomposes by the first-order reaction: C₂H₅Cl → C₂H₄ + HCl The activation energy is 249 kJ/mol, and the frequency factor is 1.6⨉10¹⁴ s^-1. What fraction of the ethyl chloride decomposes in 15 minutes at this temperature?

Video transcript

hi everyone for this problem. We're told the activation energy for a hypothetical first order reaction is 134 kg joules per mole, the frequency factor for the reaction is 2.53 times 10 to the 12 seconds inverse. At 500 kelvin, calculate what fraction of a converts into products in one minute at this temperature. So what we want to saw for here is the fraction of a. And we're told some important information here. The first is that we have a first order reaction and were given the frequency factor, were given the temperature and were given a time of one minute. And so what we're going to need to do here is use the first order integrated rate law to solve this problem. And the first order integrated rate law is Ln of our final concentration is equal to negative K. T. Plus the Ln of our initial concentration. Okay, so what we want to solve for here is a fraction of a. So we need to rearrange this equation so that we're solving for Ln of our final concentration over our initial concentration. So we need to divide both sides of this equation. Bye. The L. N. Of our initial concentration. Okay, so when we do that, we get a rearranged equation of Ln of our final concentration over Ln of our initial concentration is going to equal negative K. T. Okay, so this is what's going to give us that fraction. So we know what our T. Is our time. We're told that it's going to be one minute but we don't know what K is. Okay, so we first need to solve for K in order to be able to plug in. So in order to solve for K, we're going to need to use our we're going to need to use the Iranians equation. And that is so let's separate this here. So r K is going to equal our frequency factor times E. Raised to our negative activation energy over R. T. Okay, so let's go ahead and plug in because we have everything that we need based off what was given in the problem A represents our frequency factor and we're told that that is 2.53 times 10 to the negative 12 seconds in verse. And this is going to be multiplied by E. Which is a function on our calculator raised two. The negative activation energy over R. T. Okay, so we have our activation energy were told was 100 and kg jules. We need to convert this to jules and let's do that here really quickly. So in one killer jewel, We have 1000 jewels. So our killer jules cancels and we're left with jewels. So we get 134,000 jewels. Okay, So we get times negative 134,000. Okay, divided by R is R gas constant. Let me make this smaller. So we don't run out of room here. So we get Is divided by R which is 8. jewels per mole kelvin. So this is jules and the units here is jules over more times kelvin and our temperature is 500 Kelvin. Okay, so this is going to be multiplied by 500. Okay, so now when we saw for this, we're going to get a K of 2.53 times to the negative two seconds inverse. So now we have our K. So let's go ahead and come back to our integrated rate law and plug in what we have. So we have Ellen of our final concentration over our initial concentration equals negative K. We just saw for K. So this is going to be negative 2. times 10 to the negative two seconds inverse. Okay, so to get rid of this, L N. We need to raise both sides to E. Okay, so when we do that on the left side, we're going to just get our final concentration over. Our initial concentration is going to equal Oh sorry, this is going to be times T And we were told we're converting products in one minute. So we need to convert this two seconds. So this is 60 seconds because r K is in seconds inverse. So we need to convert that minute two seconds. Okay, and so once we raise the right side to E, we get our final concentration over our initial concentration is equal to 0.219. Okay, so now to solve for the fraction consumed, we're going to take one minus that number. Okay, so one minus zero point 219 is going to equal 0.781. Okay, so our final answer is the fraction of a that's converted into products in one minute Is equal to 0.781. This is our final answer, and that is the end of this problem. I hope this was helpful.