Gibbs Free Energy: The Relationship Between Delta G, Delta H & Delta S

[00:00:00]Gibbs free energy going to be the topic of this lesson my name is Chad and welcome to Chad's prep where my goal is to take the stress out of learning science now in addition to high school and college science prep we also offer MCAT dat and oat prep as well I'll leave a link in the description where you can find those courses now this lesson is part of my new general chemistry playlist I'll be releasing a several lessons a week throughout the rest of this school year so if you want to be notified every time I post one subscribe to the channel click the Bell notification so Gibbs free energy so symbolized by the letter G and more commonly we're going to talk about about the change in gives free energy Delta G so and Delta G is simply defined in terms of quantities we've already looked at the enthalpy change and the entropy change so this is one of the more famous equations in thermodynamics here so Delta g equals Delta H minus t Delta s now before we get there and technically that is kind of the definition of delta G it turns out it's a mathematical definition Mr Gibbs here in coming up with his definition it really was just a mathematical definition now what you've got to realize though is that when we talk about Gibbs free energy don't think of free like it doesn't cost anything it's not that kind of free it's like are you free for lunch on Friday are you available so when you think of gives free energy you want to think of energy that is available to do work energy that is free to do work so we like talking about the interaction between the system and the surroundings the system is whatever you're looking at often a chemical reaction and the surroundings is everything else in the entire universe and so turns out the system can't create its own energy if the system ends up with more free energy in this case Gibs free energy it's because it got it from somewhere it's because the surroundings gave it to it so however the system can use up its free energy again this is the energy available to do work and if the system uses up its free energy it's using it up to accomplish something and so in this case it turns out when a system uses its energy its free energy up to accomplish something that by necessity means Delta G is going to be a negative number less than zero so and in this case that's when a reaction is spontaneous so though the system can choose to use up its own energy but can't create its own energy and so when Delta G is positive that is a non-spontaneous process the the system can't create its own energy the only thing it can do is get energy from the surroundings the surroundings can give it that free energy but it can't just create it on its own and so big thing here is that when Delta G is a negative number that is the kind of reaction that is spontaneous one that actually occurs the system is using up some of its free energy to do work so but when Delta G is positive that's a non-spontaneous reaction and then finally when Delta g equals z that is a system at equilibrium so one quick thing here we'll talk about this more later but contrast Delta G versus delta G standard that little circle means standard and that means talking about the Delta g under a certain specific set of standard conditions and those standard conditions mean that all reactants and all products if they're aquous have a one molar concentration or all reactants and all products if they're gas gasas have a one atmosphere partial pressure now technically standard conditions doesn't imply a temperature but most of the time like when you look up you know thermodynamic properties say in the back of your textbook they're going to be recorded as at standard values at 298 Kelvin I.E 25 degrees Celsius and so because standard values for these thermodynamic quantities are most commonly reported at 25 degrees Celsius a lot of students Come Away with the impression that the temperature is part of those standard conditions but I'll remind you it's technically not you can have standard conditions at any temperature the big thing is it tells you what the concentrations of your reactants and products are again for aquous species all one molar concentrations for gasy species all one atmosphere partial pressures and so in this uh the last three lessons you're really going to be dealing with Delta G in this thermodynamics chapter and so I just really want to make sure you distinguish between these two that'll be important later on and sometimes as professors we don't do a good job of it either so right now we're talking about plain old Delta G here so and so we're saying here that when Delta G is negative it means that the reaction is spontaneous under whatever conditions you happen to be under that this value applies when it's positive non-spontaneous when it's at equal to zero you're at equilibrium okay so where did Mr Gibbs come up with this well uh how you know how did he come up with the idea that Delta G being negative is a spontaneous reaction well we learned from the second law of Thermodynamics what the requirement for a spontaneous reaction was we learned that Delta s system plus Delta s surroundings has got to be positive so the Delta s of the entire universe is positive for a spontaneous process that's what our second law of thermics come from so and that's actually where Gibbs began he started with this lovely expression and said well you know that's where I'll start too then so but he took this a little bit different route here so and by the way when you see system here so we technically don't have to label that one if you ever see Delta G Delta H or Delta S without a label that's automatically for the system which is often a reaction or something along these lines but if you're going to say Delta s surroundings or Delta s Universe You Better Label those so but the one that's not labeled that's the Delta s of the system well it turns out this would be a rather inconvenient way to figure out if a reaction is going to be spontaneous because measuring the Delta s of the system is one thing but measuring the Delta s of the entire rest of the universe that's kind of problematic so he said well you know turns out we could measure that indirectly though by just looking at the system so the last lesson we learned one of the definitions of Delta s was that it was Q reversible over T Q reversible over temperature and we also learned earlier in this uh earli in the course way back in chapter 5 that enthalpy was the same thing as heat for a process carried out at constant pressure and so as a result we can kind of come up with an expression for Delta s that it's equal to Delta H over T when done at constant temperature as well as constant pressure as it turns out so what we're going to do is make some substitutions here so in this case for Delta s of the surroundings we'll keep the Delta s of the system but again this is not super convenient try and measure it for the whole surrounding so what we're going to do is make some substitutions here we're going to put plus the Delta h of the surroundings over the temperature so we just substitute Delta H over t for that Delta s value of the surroundings and again this has got to be greater than zero well we can take this a step further so between the system of surroundings we're just exchanging enthalpy and so in this case if enthalpy which we most closely associate with heat uh you know leaves the system that's a negative value for Delta H and for the surroundings it would gain the exact same amount and be the exact same value but positive and so it turns out the Delta h of the system and surroundings are equal in magnitude just having the opposite sign and so you could substitute in you could say that the Delta h of the surroundings is equal to the gative of the Delta h of the system which again we don't actually have to include that label and so we can rearrange this just a little bit further and go Delta S Plus uh we'll put a negative sign in there negative Delta h of the system I'll erase that in a second just FYI is greater than zero and so once again we don't actually need to label the system so we'll take it off so but the interesting part here is that now I've got an equivalent expression to the second law of Thermodynamics of the requirement of a spontaneous reaction the Delta s minus the delta h/ T has got to be a positive number well let's rearrange this just a little bit so what we're going to do is we're going to multiply through by - t and so we're going to get - t Delta S Plus Delta H and when I multiply Z byga T I still get zero so oh but I did mess up one thing if you multiply or divide by a negative it changes the sign here so it becomes a less than sign instead of a greater than sign and so we end up with this expression right here so and if we rewrite this just the opposite way Delta H minus t Delta s is less than zero and again we derive this from the second law of Thermodynamics this is a requirement for reaction to be spontaneous something that's true for any spontaneous reaction or process and so Mr Gibbs thought this was rather convenient so that now instead of having to measure anything involving the surroundings I can just measure the eny change of the system and the entropy change of the system and not have to measure anything in the surroundings at all and come up with a requirement for reaction to be spontaneous and it turns out when this lovely quantity is a negative number less than zero that's when a reaction spontaneous said well you know I'm just going to call this combination of things Delta G and that's where Mr Gibbs came up with this idea so the derivation of this is not super important for those of you that are interested that's where it kind of comes from but you do have to know this equation Delta G is equal to Delta H minus t Delta s and sometimes I like to be factious and and shock some students I'll be like hey I talked to your professor the other day and you guys need to get higher test scores so and I TRS out I didn't talk to their Professor I didn't know their scores but they need to get higher test scores just a way of remembering that equation because that equation forgives fre energy is super important so now we know what Gibs free energy is it's a mathematical equation we know where it comes from but we know what it means when Delta G is negative a reaction is spontaneous when Delta G is positive a reaction is non-spontaneous and when Delta g equals z the reaction is at equilibrium one thing to note so if Delta G is negative in One Direction so then the opposite direction of the reaction Delta G would be the exact same number but positive and so another way of saying Delta G is positive means a reaction is non-spontaneous what also you could just say well the reverse reaction is the one that's spontaneous so you could say it that way as well just in FYI so now we got to talk about some relationships between Delta G Delta H and Delta s and some conditions for when a reaction is going to be spontaneous or not so we're going to rely back on this equation first but before we look at this mathematically uh it might be helpful to kind of talk about what the universe wants in a chemical reaction and turns out the universe wants a couple of things it wants your well one we know that free energy needs to go down well it turns out from an enthalpy perspective your best bet to have your overall free energy going down is to have your enthalpy go down as well to give off heat heat so to speak and so exothermic reactions tend to be much more likely to be spontaneous than endothermic reactions but that's only part of it so but that's one of the things the universe wants it wants Delta H to be negative wants the reaction to be exothermic and give off heat now on the other side of the coin it turns out that we learned from the second law of dyamics that one of the big things Universe wants is so for a spontaneous process the entropy change of the universe is going to be positive it's going to increase so in terms of entropy well great way to get a good start on that is just if the entropy of the system is increasing as well and so the second thing the universe wants in addition to having an exothermic reaction in a negative Delta H is it wants a positive Delta s for that reaction a positive Delta s of the system so and it turns out if you give the universe both of those things that reaction will always be spontaneous if you don't give the you know the universe either of those things like let's say instead of an exothermic reaction you have an endothermic reaction and instead of Delta s of the reaction being positive Delta s of the reaction is negative well for an endothermic reaction where Delta s of the reaction is negative that reaction is never going to be spontaneous so and then finally if you give the universe one out of the two things it wants either it's exothermic but Delta s is negative or it's endothermic but Delta s is positive well it turns out then that reaction will be conditionally spontaneous it will be spontaneous under certain conditions but you got to give the universe at least one of the two things it wants so and again the universe wants things to be lower in any energy and so things that give off heat more likely to be spontaneous so exothermic reactions so and if you looked around my office here right now you'd find out that the second law of Thermodynamics is definitely an effect it's more likely to on its own get messy rather than to get clean and the universe likes more disorder and things of this sort and so again second thing the universe wants is a Delta s that is positive for the reaction for the system all right so let's take a look at this mathematically now and before we get too far into this mathematically we got to make make sure your math skills are up to speed so my question for you is what do you get when you add a positive number to another positive number yeah you get another positive number an even bigger positive number now let's let's not go you know assume too much here what do you get when you add a negative number to another negative number great you get an even more negative number now the difficult one what happens when you add a positive number to a negative number well it depends and notice I didn't say multiply we're not multiplying here we're just adding if I add a positive number to a negative number well the result come could come out positive negative or zero depending on if one's bigger than the other if the positive number is bigger well then add of them together gets positive if the negative number was higher magnitude then it's overall going to come out negative and if they were exactly equal in magnitude then they'd add up to exactly zero these are the math skills you need and so when we take a look at this is Delta g equals Delta H minus t Delta s we're going to look at this as a plus b where a is simply Delta H but B is not simply Delta s it's this entire second term including the negative sign it's negative T Delta s and what you should realize about this second term here is that it's always going to have the opposite sign of whatever Delta s is because the temperature here it turns out has to be in kelvin and there's no such thing as a negative temperature in Kelvin it stops at zero so on the on the Kelvin scale so but you got this negative sign over here and so if Delta s is positive then this whole term is going to end up negative due to that negative sign right there and if Delta s is negative well a negative times a negative would be positive and this whole second term is going to end up positive and so now we can evaluate based on the signs for Delta H and Delta s we can look and say Well when would Delta G come out negative because again that's when the reaction would be spontaneous so let's give the universe both of the things it wants and the first thing the wants it wants an exothermic reaction and it wants the entropy change of the system of the reaction to be positive again as a contributing factor towards the entropy of the entire universe being positive all right well if we look at this then this means that a our first term is negative so and then B if Delta s is positive well again a negative times a positive times a positive means that b is going to come out negative as well and if you add a negative number to a negative number you're going to get a negative number no matter what when we say no matter what what we really really mean is no matter what value of the temperature you plug in again you can't plug in a negative number for temperature because it's on the Kelvin scale so T is either going to be zero or it's going to be a positive number but it can't be negative and so no matter what possible temperature you plug in here you're just still adding two negative numbers now it turns out at low temperatures this would be a small negative number and at higher temperatures this ends up being a large negative number but it's a negative number either way and a negative plus a negative is going to be negative period no matter what temperature we talk about so when Delta H is negative and Delta s is positive when you give the universe both of the things it wants that reaction is going to be spontaneous we say at all temperatures that's kind of how that works now what happens if you don't give the universe anything that it wants so in this case what if we say that Delta H is positive and Delta s is is negative so if we go back and take a look at this situation now we said earlier you got to give the universe at least one of the things that it wants if you don't give it either one it has no chance of being spontaneous and it turns out this is not going to be spontaneous at any temperature as we'll see so but in this case Delta H is positive so a here is a positive number and with Delta s being negative that means that negative T Delta s that whole second term well again a negative times a negative is a positive and I don't care if we're doing this at low temperatures and the second term is a small positive number or at high temperatures and the second term is a large positive number I'm adding two positive numbers in either case and Delta G is going to come out positive no matter what and if Delta G is positive that reaction is non-spontaneous so a couple different ways you could phrase this as well you could say it's nonspontaneous at all temperatures you could say that it's spontaneous at no temperatures not usually the way we say it but you could also say that it's the reverse reaction that would be spontaneous at all temperatures as well all right finally some conditional things so let's say that Delta H and Delta s are both positive or both negative and in both these cases you're giving Universe one out of the two things at once which means we're going to see some conditional uh some conditional spontaneity in these cases so let's take a look at this for a sec uh mathematically so if Delta H is positive then our first term here is positive and if Delta s is positive it means our second term here is going to be negative now if my goal is to get a spontaneous reaction then which one of these terms do I want to dominate well again I want Delta G to come out negative so I want the negative term to dominate so and it turns out that Delta H and Delta S as long as you don't change the temperature too drastically the values for Delta H and Delta s don't fluctuate a lot so the big place this equation is going to change then is by changing the value of the temperature you plug in again in kelvin and so in this case I can make this a very large negative number by putting in a very large temperature I can make this a very small negative number by plugging in a small temperature and so the key is which one has got a better chance of having Delta G come out negative well again I want the negative number to be bigger than the positive number and the negative one's the with temperature so let's make this a large temperature and so it turns out when Delta H and Delta s are both positive we like to say spontaneous at high temperatures now we have to be careful what do we mean by high temperatures well it just means over some threshold temperature just over some threshold temperature and that may be something you consider High it may be something that you don't consider high but it's not about you or it's about the reaction of Interest so let's say we took a look at say H2O solid going to H2O liquid and H2O liquid going to H2O gas now if you look at this and we'll start with the second one first and going from a liquid to a gas we're gaining moles of gas we learned earlier in the chapter that if you increase in the number of moles of gas it's definitely going to have a positive Delta s for the reaction or process so in this case that's definitely got a positive Delta s but again to boil something to go from liquid to gas definitely takes heat and so it's an endothermic process and so we've got exactly one of these scenarios we've got an endothermic process that results in an increase in entropy for the system Delta H and Delta s are both positive and you very well know that to be spontaneous to boil water at least at one atmosphere pressure you've got to have a minimum temp temperature of 100° C it's got to be over 100 anything less than 100° cus it's not going to be spontaneous anything over 100° C it will be spontaneous and so in this case when we say spontaneous at high temperatures we mean spontaneous above some threshold temperature in this case 100° C or 373 Kelvin same thing here with uh going from solid water ice to liquid water so in this case it's going to be all about 0 degrees Celsius right which maybe you don't consider high and I definitely don't consider it high living in Arizona but it's not about you or me so in this case it's all about just being above some threshold temperature and so in this case for the process of melting for this case water you've got to be above 0 degrees cus not below any temperature above that would be considered a high temperature for the melting of water and any temperature below 0 degrees Celsius would be considered a low temperature for melting of water so that's kind of how that works so I just want to make sure when you see the word high here you don't think about what you personally consider hot or cold all right if we go go uh one step further by the way one one thing to note here I want to clarify that high temperature if you were asked to calculate what at what temperature that is well again the key is you know if we looked at like say again the boiling of water so if you plug in the Delta H for uh vaporization and the Delta s for VAP vaporization of water well it turns out if you plug in any value of T higher than 373 Kelvin again these got to be put in kelvin so higher than 100° C which is 373 Kelvin then Delta G is going to come out negative if you plug in any temperature lower than 373 Kelvin in for T then Delta G would come out positive and if you plugged in exactly 373 Kelvin that's when Delta G would come out to zero and so often times you're asked to find that threshold temperature and so to find that threshold temperature you just set Delta G equal to zero and then plug in the value for Delta H and Delta s and then solve for that temperature it comes out in kelvin now we'll see some examples of this later you just got to be careful on your units because Delta H is usually given in kilog per mole but Delta s's are usually a lot smaller than Delta H's and those are usually given in Jewels per mole Kelvin and so with this being in kilj per mole but this only in Jews per mole Kelvin you either got to make them both Jewels or both kogs before you start doing your calculation all right so last scenario here where Delta H and Delta s are both negative so if Delta H is negative well that means definitely our first term is simply negative but if Delta s is negative again a negative time a negative means your second term is positive and again that second term here that whole negative T Delta s is always going to have the opposite sign of whatever just plain old Delta s has and so in this case again we're adding two numbers one's negative one's positive if my goal is to come out with an overall negative number that way Delta G comes out negative and the reaction is spontaneous well then I want the negative term to have the larger magnitude well again I can't really change the value of delta H in any significant way but the second term has temperature in it and in this case I don't want to make it as big as possible I want to make it as small as possible that way the positive term will be smaller than the negative term and so in this case I want to limit the size of it by lowering the value of T I plug in and so here we'd say that the reaction is spontaneous at low temperatures and again it's not what you or I would necessarily consider low it's just mean below some threshold temperature and so instead of melting ice what if I wanted to freeze liquid water well freezing liquid water is spontaneous below 0° C it's nonspontaneous above 0° CSUS that's kind of how it works so low in this case would just mean below zero well what if I wanted steam to condense into liquid water well again steam will condense into liquid water if you're below 100° C so would 90° cus be considered a low temperature in this case yes now do you personally consider 90 degrees Celsius a low temperature no so again it's all about is it below the threshold temperature for the given reaction you're looking at cool you've got to know these four situations and when that reaction is going to be spontaneous either at all temps no temps High temps or low temps now a lot of students simply are just going to memorize this chart and if you want to go that route fantastic so whereas some students will actually work out the math every time in such fashion and I I personally do it that way myself um well truth be told I kind of have it memorized now as well so but I personally like that way so but I usually get about half students working out the math half the students just memorizing this chart whichever one makes more sense for you personally go with it but at the end of the day you are definitely on the hook for knowing given the sign of Delta H and Delta s under what conditions that reaction is going to be spontaneous in terms of temperature so the first problem we're going to look at is you're given a reaction here N2 plus 3h2 going to 2 NH3 you're given the Delta H for this reaction and then you are asked whether whether or not this reaction is spontaneous at all temperatures no temperatures high temperatures or low temperatures and so obviously you're just supposed to match it up with one of these four scenarios well right off the bat your told Delta H is negative it's an exothermic reaction you know that for sure you're giving the universe one of the two things at once the question is what is Delta s because unfortunately Delta s was not provided however in the last lesson we learned how to predict the sign for Delta s and the single most important factor was change in moles of gas and we're definitely going from one two 3 four these are both gases so four moles of gas ending up as two moles of gas that's an overall decrease in the number of moles of gas and therefore a definite decrease in entropy that tells us that Delta s is negative and so as a result we've got Delta H and Delta s both being negative and that reaction is definitely only going to be spontaneous at low temperatures and again if you want to look at this kind of mathematically you go back here and say okay if Delta H and Delta s are both negative well then my first term is negative but a negative time a negative means my second term is positive and if I want the negative term to win so that Delta G comes out negative then I want to limit the size of the second term which means make that temperature small and so that's where we get this idea of it being spontaneous at low temperatures all right that is the first example of problem let's do a little bit of math here as well okay so first thing I want to look at is Phase changes with regard to uh Delta G Delta H and Delta s and so it turns out when you are at a phase change temperature like the boiling point or the melting point AK melting point same thing as the freezing point FYI uh it turns out you have two phases in equilibrium at the boiling point you have liquid and gas in equilibrium at the freezing or melting point you have solid and liquid in equilibrium and so having them in equilibrium means that Delta g equals zero at the phase change temperature well if we look at what that implies then we could say that zero equal Delta H minus t Delta s so but again in this case let's say we're doing this for this problem we're going to want we're given the Delta h of vaporization and the Delta s of vaporization that means the temperature at which this is going to be true is going to be the boiling temperature and then we can rearrange this and notice we can just take this Delta H vaporization on this side and on this side we'll add T * the Delta s vaporization and we'll divide through by that boiling temperature and get that the Delta s vaporization equals the Delta h of vaporization over the boiling temperature so and that's where this kind of comes from so and you can do it again for any phase change so any phase change Delta g equals z and we can derive expressions like so so and as a result you've now got two equations they've got three variables and if I give you two of them you should be able to calculate the third which is what we'll do in this next problem it says for water Delta h vaporization is 4.68 K per mole Delta s vaporization is 109 Jew per Kelvin mole what is the boiling temperature of water so we want to get this boiling temperature so in this case we can rearrange this and get that the boiling temperature so we'll move this up and then we'll divide through by the Delta s vaporization but it's going to equal the Delta h of vaporization divided the Delta s of vaporization and again both these values were provided now we got got to be a little bit careful Delta H vaporization was given as 4.68 K per mole the Delta s was given as 109 Jew per Kelvin mole so whether it's mole Jews per mole per Kelvin or Jews per Kelvin mole same diff so and then we can calculate that and we'll see that that's where the Kelvin part is going to come from so we need all the other units to cancel though and the moles part that's great but kils and jewels those aren't going to cancel each other out and so here's again one of those places you got to be careful in fact Delta G and Delta H are most commonly reported in kles because they're significantly larger than Delta s which is only reported in Jewels per Kelvin so it's kind of like if I said hey how tall are you and you said you know six foot tall and I said no no no how many miles tall are you and you just kind of looked at me funny because you probably have no idea how many miles tall you are some you know 0.0 five or something miles taller maybe probably smaller than that actually so but we don't know we don't we would never measure our height in miles that would be the most inconvenient unit ever well same thing it turns out for Delta s Delta s we wouldn't use kogs because here instead of being 109 it would have been 0.19 and so it's much easier to instead of talking about fractions to talk about whole numbers and that's why for Delta s which is usually much smaller than Delta G and Delta H we'll use Jews per mole Kelvin instead of kilog but it does mean that when when you're starting to do some calculations you need to check your units put everything in KJ or everything in Jewels it's your choice it'll work out the same either way but you need to go there so we need to change some things here and personally I like changing kles to jewels instead of instead of 46 40.6 8 we're going to have 4,680 let's make that eight look a little better and now that's than Jewels rather than kles and now we're ready to do a little bit of math let me grab my calculator all right so 4,680 divided by 109 is going to give us why would you have ever done it this way you should have been like I know that one so but odds are they're going to give this you know kind of a calculation for something other than water and I'm I'm guessing that you haven't memorized the boiling temperatures of every known liquid right so but I wanted to work one out for one that you did know just to show you how it worked out and again the big things here is that when you solve for your temperature it's going to come out in kelvin so your Delta H and Delta s have to either both be in Jewels or both in Kill The Jewels for this to work out properly so but often times Professor loves to get you this far not only could you have made a mistake and not making your units match but then once you get the right answer they might put 373 de c as one of the answer choices on the test but it's not 373 Celsius it's 373.15 and then 100 degrees celsus so so be careful a couple different ways professors can kind of give you some good detractors to throw you off on just such a calculation now if you found this lesson helpful the thumbs up button and a comment are the best things you can do to support the channel if you're looking for practice for your general chemistry course take a look at my general chemistry Master course it includes over 1,200 practice questions I'll leave a link in the description free trial is available happy studying