State Postulate in Thermodynamics: Intensive & Extensive Properties Explained

okay so far our work done to define properties of a system has led us down a bit of a path

so we've looked at what different types of of condition we're looking at

the geometric state versus the thermodynamic state we've looked at different cases of equilibrium

and the properties that drive them so we've got mechanical equilibrium driven by pressure differences we've got

thermal equilibrium driven by temperature differences before we start talking specifically

about how to define properties we have a couple of other things to do one of them is to look at

something called reversible work modes and we introduced this concept of work and this is this comes up again

this was the work done to compress a gas and so this actually represents a reversible work node

it's called pdv work we're going to say that a lot in this course and we also said that in in very general

sense the amount of work done is equal to some generalized force times some generalized

distance and now we can look at this a little more closely thinking in terms of properties so when we think

about reversible work modes the displacements are actually properties if it's a change in the

position it's a geometric property or change in volume or something like that the forces

are properties associated with equilibrium so they're derived from say pressure pressure is a property

that's that's um associated with mechanical equilibrium

other forms of reversible work modes other than the pdv work are things like electrostatic

work magnetic work work done against a gravitational field

and there's a couple of other examples in your book as well and when we say reversible reversible is

a word that we use a lot in thermodynamics and as we progress in the course we're

going to get more and more refined definitions of what reversible means for now we just say that the work

is reversible in the sense that the work done on the system in one direction so think about the compression of the gas

we're squishing it so we're doing work on it can ideally be returned in the reverse

direction so if we let go of the piston the system will expand against it and give us everything back that we just put

in now this would only happen if the whole system was frictionless and so forth

which we've said it was to this point what that also means if we think more in terms of a cycle

a system would return which in to its initial state when a cycle is complete so you think

about a four-stroke engine for instance a cycle compression expansion exhaust power and so forth

so let's keep moving with properties of a system so i've listed here some well-known

properties that we've already talked about in in these lectures the pressure the

temperature the volume the internal energy the density the mass and there's going to be other properties

that we introduce as we move through the course and start developing the full forms of

the laws of thermodynamics what we want to do now is we want to classify

properties into two categories i mean we've already talked about for instance geometric properties versus

thermodynamic properties this is a little different this is a classification of properties

and the two types are intensive properties and extensive properties and the

definition is that an intensive property does not depend on the extent of the system

well an extensive property does depend on the extent of the system so if we just look at these properties

that we've listed up here and kind of categorize them as intensive or extensive let's think of

the easy one so something like the volume actually tells you how big something is

so it definitely tells you about the extent of the system so volume belongs as an extensive

property the mass is something else that tells you about the extent of the system

it's this big it weighs this much right so that's another property

the internal energy in joules or kilojoules tells you how much energy is in the

system so it also has something to do with the extent of the system so the uppercase u internal energy is

something about the extent of the system now let's look at the other ones the pressure

the pressure is in newtons per square meter or pascals or kilopascals or many other units

but me just telling you the pressure doesn't tell you how big the system is if i told you the pressure is you know

210 kilopascals you don't know whether i'm talking about a tire or a tank of compressed air or you know

some some place under a certain amount of water so the pressure itself doesn't define

the extent so it belongs in the intensive category same thing with temperature if i tell

you the temperature is 25 degrees you don't know whether i'm talking about this room

or what's outside or the water in my pool or something else so just knowing the temperature doesn't

tell you how big a system is it just tells you about something about the system

and the same thing with density okay the density tells you how much something weighs per unit of

volume right kilograms per cubic meter so it doesn't tell you how big the system is

it just tells you how dense the system is now there's an easy conversion

between intensive and extensive properties and that is simply the division by mass

and we're going to introduce another property here actually two we're going to look at this

new property called specific volume and the specific volume is simply the volume divided by the mass

so the specific volume is in units of meters cube per kilogram so it's effectively the inverse of the density

right the density is in kilograms per cubic meter specific volume is meters cubed per

kilogram the other one is the internal energy so when we talked about internal energy we

were talking about joules or kilojoules of internal energy division by mass gives us technically

the specific internal energy which we still just call the internal energy so we

we distinguish these two by looking at total internal energy versus internal energy which is a property that

you can that you can graduate and put in a chart the reason why we want to do this is because

for us to to uh kind of classify properties and and to graduate tables with the property values in it you can't

have the system involved right because otherwise there would be an infinite number of

properties because they depend on the extent of a system if properties don't depend on the extent

of the system then you can characterize them as you know functions of pressure and

temperature for instance the density of a gas can be characterized in terms of pressure and

density specific volume of a gas similarly um so this is the

the reason why we need to kind of categorize these two types of properties is to kind of prepare to um document

properties of substances in in different states

so the next question we want to ask ourselves in terms of properties is how many thermodynamic properties

does it take to fix a system i've worded it a little differently how many thermodynamic properties are

independent now we need to know this to characterize the state the question we're really asking is how many things

do you have to pin down before you can't change anything else about the system

so let's look at a really simple kind of mathematical problem in a very pragmatic way because that's the way we want to

study the thermodynamics problem so let's consider this system s which is a function of x

y z and t so four independent variables and let's add the constraints that y is a function of t

and z is a function of x and t so the question we want to ask ourselves is how many variables does s really depend

on another way of saying this is how many of them do we have to fix

before nothing else about the system can change so in a very pragmatic way let's just

make some guesses and then and then go through the system and see whether we've actually fixed the

whole system so let's guess that s is a function of one independent variable and we'll just pick

t now does this alone fix the whole system from changing well

it fixes y because y is a function of t only so that's fine but only fixing t does not fix z because you can change x

to vary z or change z to vary x so the answer to this one is no one variable is not enough

to fix this system from moving so let's make another guess let's guess that the system is a function of

two independent variables and this time we'll pick z and t so once again let's go through the

system y is fixed because we fixed t it's actually the second point

if we fix z and t you can no longer change x so it's actually fixed

that so by fixing z and t x can't change y can't change z is fixed t is fixed so that means that

the whole system is fixed so we can't actually change anything else

so this is a yes so this system therefore depends on two independent variables and

it turns out that you don't have to pick the ones i picked i picked z and t but any combination of z and t x and t

x and y fix the system and i'll let you go through those other examples now the reason this is important is

because when you're characterizing properties and you're tabulating properties and

you're measuring things about a system you want to know how many things you have to measure before you can

understand everything else about the system in other words suppose you're you're after

some difficult property like internal energy it doesn't necessarily mean you have to

measure internal energy it's a lot easier to measure pressure and temperature for instance

so if you're after the internal energy and you know the system is only a function of two variables you pick the

easy ones to measure and then you just look up the other properties which is something we're

going to be doing now what i want to do is i want to take this exact example

and i want to apply this to a thermodynamic system to understand how many properties we have to define

or that have to be ended that have to be used to characterize the state so let's move on to the thermodynamic

system using the same kind of approach that we took in that very simple mathematical system

we considered so what i want to consider is a very basic piston cylinder device

containing f fluid let's call it a compressible fluid and so the system is captured inside of

this dashed line and it's just of mass m we can do heat transfer and work on this system in

general um so what we want to do is is we first want to understand that

that the two properties we're going to be after are independent of one another okay so we're going to look at the

internal energy and the specific volume they're intensive properties and we just want to see that they're

independent so how do we do that well let's fix one of them so if we fix v we could do that by

pinning this thing and then trying to change u okay so if we fix v by pinning it it

pins the volume down since the mass is constant that means that v is fixed no matter

what else we do and the question is can we still change you well the answer is yes

because we're allowed to to add heat or remove heat uh through the system boundary and that

has a direct effect on the internal energy of the system so u can be changed by del q if we fix v

now the other question is if we fix u can v change and the way we have to think about this is by looking at the

first loss statement for this that's the easiest way to think about it and the first loss statement is del q

minus del w according to this convention is equal to delta e

but if the system is stationary so no changes in kinetic or potential energy then this

amounts to delta u which we could also write as knowing this difference between

intensive and extents we could write this as m delta little u so the question is

if we fix internal energy can we still change heat and the answer is actually yes because

what we're doing is we're setting the right hand side of this to zero because that's the delta u and that

still says that as long as del q minus del w is equal to zero

then we haven't changed u so in other words if we compress it and allow an exactly equivalent amount

of heat to leave as work was put in then delty then then the internal energy hasn't changed

and vice versa we allow it to expand but we add heat it would amount to the same thing so

by doing these two simple observations we understand that these two intensive properties

internal energy and specific volume are independent of one another so now we can move on with um

is is fixing these two properties enough to fix the entire system so now let's go back to this problem so

what we're going to check is if we fix u and v can we change any other property of the system the

properties we want to think about are things like pressure and temperature so question one can we change p

well what are the ways that you would think about to change the pressure of this system

you could think to squeeze the fluid or to try and pull it apart so expand or contract the fluid

well we can't do that because if b is fixed v would have to drop for us to squeeze

the fluid so the answer is no we can't we can't compress the fluid or expand the fluid

we can't squeeze it another way of changing p would be to heat it up well can we heat it up

no because if we fixed u heating the fluid causes a change in you so we also can't change the pressure by

adding heat that's a no another way we could do and i didn't show it in this picture but we could

stir the fluid because that's another way of adding work and then if we let the system come to

rest that work through the viscosity would become internal energy but if we do that

that also results in a rise in internal energy so we can't do that either because we fix both of these so all of

these three ways that you could imagine to change the pressure of the fluid inside the cylinder aren't

allowed if we fixed the internal energy in the specific volume

what about temperature can we change the temperature if we fix these two properties and the

answer is no again because the way you would change the temperature is very similar to the way you change the

pressure you'd squeeze it because it would heat up you could simply add heat you could stir

it but all of these things would result in a change in one of these properties that

we're holding fixed so in fact similar to the mathematical problem and

i kind of intentionally picked the two variables if we fix two independent

intensive properties then we have basically made it so that no other property of the system can change

the density can't change the pressure can't change the temperature can't change

and also other properties that we haven't even talked about yet but uh the hope is that by understanding

kind of taking this pragmatic approach the way we've done it is we first picked two properties two

intensive properties we have concluded that they are independent of one another

because fixing one doesn't fix the other but if we fix them both then you can't change anything else so

that that's the analysis we've done here this is very important and it's going to lead to a statement

that i'm going to make in the next short video so continuing where we left off um

i want to make a statement actually about the number of properties that you have to fix in order

to hold the rest of the system or substance at the same state so i'm going to make a

few statements here leading to the probably the most important thing we've done in thermodynamics today

so statement one says if there are n reversible work modes and remember a few lectures ago we defined a reversible

work mode as um you know a dell w is force times some distance we've got

pdd is a reversible work mode plus others associated with magnetic fields of

electric fields and so on so if there are n reversible work modes defined by this so i've got

work mode one work mode 2 all the way down to work mode n then fixing all of the x's

which you'll recall are associated with properties so fixing all of the x's x1 x2 through

xn and u which is one other independent property fixes the thermodynamic state

okay now we need to go a little bit further before we can make the important statement

so statement two defines this new thing called a simple compressible substance or

system an scs so a simple compressible substance

is one for which pdv is the only relevant reversible work mode so there are many

we know what they are now but a simple compressible substance and it's a substance that we're going to

spend all of our time in the first thermodynamics course on simple compressible substance is one for

which pdv is the only relevant reversible work mode

so what this means is that there is only one of these so we fix x1 which in our case is

v right and that's what we've done so fixing v and u we can look up all of the other

properties everything else about the system is fixed we showed that in the last

uh short lecture so in other words the pressure is defined if we understand what the

internal energy and the specific volume are we can look up the pressure we can look

up the temperature alternatively if we know the temperature in the specific volume we can look up

the internal energy and there's other combinations knowing p and t and so forth

and this leads us to probably the most important statement that we can make so far and this is concerning properties

and this is going to kind of lead us through the next bunch of lectures where we actually get

into the specific details about the properties of substances and this is called the state postulate

if i was lecturing to the group in a normal sense i would ask this question

at least every other lecture i walk into and i would make somebody tell me what the answer is

the state postulate says that the state of a simple compressible substance is completely defined by two

independent intensive properties right so there's four really important things it's a simple compressible substance an

scs and it's completely defined by two independent intensive properties so when

i ask you this on a test and i will ask you on every test to tell me what the state postulate is you're going to

tell me in detail showing these four things because otherwise it's wrong you have to include

all four things so this this kind of gets us to the end of the little the little pieces that we

have to study so we've looked at different types of properties we've categorized the properties we've talked

about equilibrium right mechanical equilibrium thermal equilibrium we've related that to

reversible work modes now we've looked at you know the the exercise of how to understand what you

have to pin down so that nothing else can move from this we've actually made some statements

that are lead to a postulate so the one about the numbers of reversible work modes the

simple compressible substance and now this very very important statement of the state postulate

so the next set of lectures we're actually going to start studying water we're going to take a long time to study

the properties of water because water is a really interesting substance

and exploiting the properties of water isn't possible unless you know the properties of water

which is is actually a very complicated thing and this is the next subject we're going to tackle