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because the dehydration reaction is not observed to occur at 298 Kelvin the student claims that the reaction has an equilibrium constant less than one point zero zero at 298 Kelvin do the thermodynamic data for the reaction support the students claim justify your answer including a calculation of standard or change in standard Gibbs free energy at 298 Kelvin for the reaction so let's first follow us for a view what the students claim is the student claims that since the the reaction is not observed at 298 Kelvin that at that temperature the the the reaction the student claims that the reaction has an equilibrium constant less than 1.0 at 298 Kelvin that's their claim and if these idea if the ideas of equilibrium constant or Gibbs free energy are completely foreign to you I encourage you to review the videos on Khan Academy on Gibbs free energy and equilibrium constants if they're familiar to you but you're like okay I don't know all the formulas that might connect the information that's given in the problem and how to calculate Delta G and maybe how do we go from Delta G to equilibrium constants and figuring out whether equilibrium constants going to be greater than or less than 1 or equal to 1 the good thing is is that they give you all the formulas that you need it's on the first page of the free response section you just have to understand which ones are applicable and and what's actually going on with those equations so let's first of all think about whether we can what what let's think about whether we can calculate Delta G with the information that they've given and if you go to the first page on one of the formulas they give you and you might even remember on you know how do you figure out whether a reaction is spontaneous or not and how you could calculate Delta G based on the temperature the change in entropy and the change in Delta P the you get you have this formula right over here and I literally just copied and pasted it from what they give you when you take the test and so they give us the change in enthalpy for the reaction they give us the temperature 298 Kelvin they give us the change in entropy for the reaction this is the change in standard entropy change in standard enthalpy for the reaction and so using those we can figure out what Delta G is going to be and then we can say okay is it consistent is it is it greater than is it greater than zero which is consistent with the reaction not being spontaneous which seems to be what's observed and then how can we go from that Delta G to the equilibrium constant well we they also tell us that Delta G is equal to this second thing and this connects Delta G and the equilibrium constant and we know some things about R&T in fact we know exactly what R and T are if we need them so let's go ahead and and apply apply these equations right over here so you have actually let me just write all the information we have first so what is our change in standard enthalpy I have to figure that out we're gonna have to figure out what our change in standard entropy is going to be we know what the temperature is we know that that is 298 Kelvin and we could worry about our in a second if if we need to so up here when they gave us the reaction they gave us our change in standard enthalpy and our change in standard entropy at 298 Kelvin this is very convenient so our change in enthalpy is 45.5 kilojoules per mole our change in standard entropy is 126 joules per Kelvin mole so this is very interesting this is given in kilojoules this is given in joules and so we want to make sure that we're being consistent so let's let's just make sure everything is in joules so I'm gonna write this this thing is the same thing as 45,500 joules per mole for this reaction so let me write this down this is 45,500 this is 126 joules per Kelvin mole so this is this right over here is 45,500 joules per mole this is 126 joules per Kelvin mole and now we can figure out what the Delta G is going to be the Delta G is going to be equal to well our change in enthalpy is 45,500 joules per mole minus our temperature 298 Kelvin times are change in entropy times 126 joules per per Kelvin per Kelvin mole and notice the Kelvin and then the Kelvin those should both be capital Kay's it's a capital K cancel out and the unit's here gonna be joules per mole the unit's here gonna be joules per mole and then we just figure out the difference so let's let's let's do this I'm gonna get my let me get my calculator out and so this is going to be this is going to be when we multiply these two first you're gonna have 298 times 126 is equal to that and let's see I'm gonna subtract this from 45,500 so let me just make it a negative and add it to 40 5500 is going to be equal to is going to be equal to seven thousand nine hundred and fifty-two and if we want to stay and we want to stay consistent with how many significant digits or significant figures we have and it looks like it's pretty consistently three significant figures so we want three significant figures here so we could write seven thousand nine hundred roughly seven thousand nine hundred and fifty so our Delta G is approximately 7,950 joules per mole and the fact that this is greater than zero tells us that this is not going to be spontaneous at that temperature so it's already consistent with our observations and that's always a good reality check is are the things that you're seeing consistent with what the what the what the question is describing so Delta G greater than zero consistent consistent with reaction reaction being or not being spontaneous not being spontaneous spontaneous at 298 Kelvin and we in the videos on Khan Academy on Gibbs free energy we go into a lot more detail on what this is but one way to think about it is this is the energy the change in energy that's of to do work and so if you have more if you if your Gibbs free energy increases over the course of the reaction that means the products have more energy to do work and that means you have to put work into it in order for the reaction to actually proceed if your Delta G is negative that means your products have less energy to do work than your reactants which means that it can release that energy and it can do work and it can be spontaneous so this one it's greater than zero so it's not going to be spontaneous but that doesn't answer the question for us we want to we want to validate the claim that the reaction has an equilibrium constant less than 1 at 298 Kelvin and lucky for us they also give us the formula that ties our Delta G to our equilibrium constant and we know the other things we know are and we know T and we might not actually even have to think about them because we don't they're not even telling us to calculate the equilibrium constant they're just saying we'll validate that it's going to be less than 1 and so if we took 7,950 joules per mole is equal to negative RT times the natural log of our equilibrium constant well solve for the equilibrium constant let's see if I divide both sides by negative RT negative R T I'm going to be I'm going to get the natural log let me just write it this way the natural log of my equilibrium constant is going to be equal to 7,950 joules per mole over negative negative RT or you could say e this is just what power do I raise e to to get K or so you could say e to the negative 7,950 joules per mole over RT is going to be equal to K and we could actually calculate what this is we know which are to use we're dealing with jewels and moles so it would be this first gas constant right over here 8 point three one four joules per mole Kelvin but we actually don't even have to do that because we just have to validate that K is going to be less than one what happens if you raise e to a negative exponent and this is this is going to be a negative exponent this is positive that is positive this is positive we know it's 298 Kelvin positive positive so my entire exponent is going to be negative so negative so negative so you could say that K equals e to negative negative number I could just write it's kind of weird right the negative and then the negative number e to a negative negative number which must be less than one remember if if the exponent is 0 e to the 0 power is 1 e to anything positive is going to be greater than 1 and e to anything negative is going to be less than 1 let me be careful not less than 0 you actually can't get to less than 0 it's going to be less than less than 1 and so this by itself already validates the students claim if you want to go further you could just calculate that you could just say K is equal to e to the negative 7,950 joules per mole over over R which is what 8.314 joules per mole Kelvin 8.314 8.314 joules per mole Kelvin times 298 Kelvin those cancel out and joules per mole divided by joules per mole those cancel out and so you would get your number and actually let's just calculate it just just just for kicks just to really feel good about it and you can see this is going to be dimensionless and equilibrium constants are dimensionless and so we are going to get this is a this is kind of fun so let's see let's let's do this denominator first if you have 8.314 times 298 times 298 that's going to be equal to that and now we're dividing by that so let me take the reciprocal of that and multiply it by seven thousand nine hundred fifty-seven thousand nine hundred and fifty is equal to that now we want to raise e to the negative of that so let's make that negative and now let's raise e to that power so let's see who can raise e to that power so we just press that and there you have it this is approximately zero point four four zero I guess you could say so this is or four zero well let's just say four zero four so this is approximately approximately zero point four zero four approximately zero point we'd already forgot the number I have a bad memory zero point zero four zero four zero point zero four zero four and the whole rules of significant figures get a little bit trickier when you're starting to deal with exponents like this we're not gonna and they don't ask us to actually calculate the exact value but they're hopefully this makes you at least appreciate that the equal equilibrium constant is for sure is less than one