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Concept #1: The Second Law of Thermodynamics for an Ideal Gas

**Transcript**

Hey guys, in this video we're going to talk about the second law of thermodynamics as it specifically applies to an ideal gas like pretty much everything we do we want to talk in general and we want to apply it specifically on ideal gases because that's mainly what we deal with in thermodynamics let's get to it. Remember the entropy is a state function it only depends on the state of the gas and there are a bunch of consequences to that this means that a change in entropy most importantly is path independent it only depends on the change in the state of the gas so when you go from one state to another from some initial state to some final state it doesn't matter how you get there because all that matters is what the initial state is and what the final state is so the path doesn't matter. The state of an ideal gas is determined by two of the following three properties pressure volume or temperature any two of them will define the state property it's common to represent the state of a gas as a combination of pressure and volume right because we use PV diagrams so commonly on a PV diagram for a particular combination of pressure and volume you define state but for entropy it's actually more useful to represent the state of an ideal gas with temperature and volume just because that's how most people represent this equation that we're going to get to in a second we can use the first law of thermodynamics right that delta U equals Q plus W to find the change in entropy for an ideal gas it will be written in terms of the state given by the temperature and the pressure now how to actually convert delta U equals Q plus W that first law into this representation of the second law I'm not going to get into your book will probably do it your professor might do it if you do a lot of derivations in class but it's not worth doing here the final result is that the number sorry the change in entropy for the gas is the moles of the gas times the specific heat at constant volume times the logarithm of the final temperature divided by the initial temperature plus the number of moles times the ideal gas constant times a the logarithm of the final volume divided by the initial volume right and the second law of thermodynamics says this has to be greater than or equal to 0.

If the change in entropy of the gas is greater than 0 the process can occur on its own right if the change in entropy for the gas is less than zero the process cannot occur on its own I'm not saying you cannot occur at all because the gas is not the only thing that a system can be made up of right a gas can be part of an engine and that engine can be a part of a hot or cold reservoir and those reservoirs can influence a change in entropy for the system but it cannot occur without help. There needs to be heat flow from a hot reservoir to a cold reservoir that's always going to lead to a change in entropy of the reservoirs greater than 0 this is something that we addressed when we first introduced entropy that whenever heat flows from a hot reservoir to a cold reservoir the change entropy for those reservoirs always greater than 0 no matter the temperatures if the heat flow is large enough then when you add the change in entropy of a gas to the change in entropy of a reservoir you're going to get a number greater than 0 and the second will be satisfied right this is the basic idea of how an engine works that even though. The change in entropy of the gas is negative yeah even though the change in entropy for the gas is negative that the gas decreases it's disorder that the gas decreases it's entropy which seems to violate the second law It actually doesn't the reason is that the total change in entropy for the system which is a combination of both the change in entropy of the gas and the change in entropy the reservoirs that does increase that does become greater than the initial state so the second law is satisfied even though the change in entropy for the gas is negative this is the basic idea with heat engines that you can overcome certain processes that won't occur on their own by adding in reservoirs.

Let's do an example 4 moles of a monoatomic ideal gas is expanded isothermally from some initial volume to some final volume what's the change in entropy does it satisfy the second law of thermodynamics right the change in entropy for a gas is always going to be the number moles times the specific heat at constant volume times a logarithm of the final temperature over the initial temperature plus the number of moles times R the ideal gas constant times the logarithm of the final volume over the initial volume, something very very important to note is that the logarithm of one is always zero that means that if temperature final equals temperature initial. Then you get the logarithm of one which is 0. That's true for an isothermal process right so this whole term is 0 because the final temperature equals the initial temperature so you're going to get a logarithm of 1 so all we're left with is this term so we have 4 moles the ideal gas constant is 8.314 and we take the logarithm of the final volume 0.008 divided by the initial volume 0.005 plugging this into a calculator we see that this is 15.6 Joules per Kelvin is the second law of thermodynamics satisfied, yes just by the gas alone the second law of thermodynamics now is satisfied and this process can occur without any help without any heat flow from reservoirs this process can occur on its own. Alright guys that wraps up this discussion on the second law of thermodynamics and how it specifically applies to ideal gases. Thanks for watching you guys.

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Concept #1: The Second Law of Thermodynamics for an Ideal Gas

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