Understanding Vertical Displacement and Time Taken in Thermodynamics

let's say you and your friend are in front of this building and both of you want to get from the ground floor to the

top floor which I've marked by points A and B and let's say one of you decides to take the lift and the other person

takes the stairs so in this situation let's say we want to calculate two values and these are we want to know

what is the vertical displacement as you both go from A to B using different paths and we want to know how much time

is taken to go from A to B in both of these cases that is by the lift and by the stairs so in the first case it

doesn't really matter if you take the lift or if you take the stairs the vertical displacement is going to be

same because what we're doing is if we think of this ground as our point of origin we are simply calculating the

difference that is y b minus y a and in both cases the displacement is the same you can take the lift or you can take

the stairs but you both are starting from the same initial position and you're both ending at the same final

position so if you think of vertical displacement as a function it is only depending on the initial and the final

States and it does not depend on the path that you have taken to get there now let's look at the second calculation

which is the time taken and assuming this is not a particularly slow lift we know that the time taken by the lift

will be less than the time taken through the stairs so here both you and your friends started from the same initial

position went to the same final position but the time taken was different because you took different paths to get from A

to B and so if you think of both of these quantities as functions the first one depends only on the initial and the

final position whereas the second one is dependent on the path that you took to get from A to B now let's see how we can

use this idea in the context of thermodynamics in the situation that we saw just before

we had an initial and a final position that we called A and B and if we are studying a thermodynamic system we need

some ways to Define these initial and final positions so for that we use some thermodynamic variables which are things

like pressure temperature volume and number of moles so these are some of the most commonly used thermodynamic

variables and how we use them to describe the state of a system is that we can say that the system went from

some initial state with pressure P1 temperature T1 volume V1 and N1 number of moles to a state 2 where the pressure

then became P2 the temperature was T2 the volume changed to V2 and the number of moles was N2 so these variables

describe the initial State and these describe the final state of the system also it's not necessary that all of

these have to be described to Define the initial State we could just describe let's say the initial volume and number

of moles and the final volume and the number of moles or even just one of these but the point is we could be using

a combination of any of these variables to describe the initial state of the system and the final state of the system

which was like identifying the points A and B in the example that we saw before and just like we saw before we can also

have thermodynamic functions which are either State functions or path functions the difference being that the state

functions will depend only on the initial and the final State and the path functions depend on the exact path that

we take to get from State 1 to 2. let's go through them one by one if we think of a system which is going from this

initial state to this final State and let's say we know that the initial pressure was P1 and the final pressure

was P2 if we look at the change in pressure which is the Delta P that will only depend on the final and initial

pressures because the Delta P will be P2 minus P1 so this is an example of a state function similarly we can also

look at the change in temperature or the change in volume or number of moles and all of these are state functions another

state function that we have is internal energy which is denoted by a u and if you think of a box with some gas

molecules in it the internal energy is a sum of different energies like the rotational energy of the molecules or

the bond energies so this is one catch-all term that we use for all these different energies then there is also

entropy which is denoted by S an entropy is a measure of how the order in a system changes as it goes from say State

1 to state 2. state functions can also be a combination of other state functions like we have enthalpy which is

internal NG plus pressure times volume or Gibbs free energy which is enthalpy minus temperature times entropy and

don't worry if all of this seems a little bit unfamiliar right now because you learn more about them as we go

through more thermodynamics but the point is all of these quantities the internal energy entropy enthalpy Gibbs

free energy all of these are State functions which only depend on the initial and the final State and on the

other the side we have path functions the most common examples of path functions are things like work and heat

which is basically the work done by a system or the heat added or removed from the system so to see how these are path

functions let's take an example let's say we are going from some initial State P1 V1 to some final state which is p to

V2 so for this change if I draw a PV diagram which is basically a graph of pressure versus volume it will look

something like this so here we have our initial pressure and volume and this point corresponds to the initial State

and this point is the final state where the pressure has now become P2 and the volume is V2 so as we go from 1 to 2 you

can see that the pressure and volume both are increasing and in this situation I can think of two ways to get

from one to two so let's say I first start at this volume V1 and I keep the volume constant and I only increase the

pressure so I go from here to Here and After increasing the pressure to P2 I then increase the volume from V1 to V2

so what I've done is I've broken this into two steps so apart from this another way to get from 1 to 2 would be

to First increase the volume from V1 to V2 and then increase the pressure in the Second Step via both these parts we got

from our initial state to our final state but we took different approaches to get there so now if we were to

calculate the work done on the system as we go from 1 to 2 it will depend on the path that we choose because we know that

the work done in this case where we have a PV diagram is the area under the curve and we'll see later how we can derive

this but for now let's just take it as a fact that we know that the area under a PV diagram is the work done so in case

of the yellow path the area under the curve in this case will be this portion which I have shaded in yellow here let

me just separate this out and now in the second path if we were to calculate the work done it will be the

area under this green curve which is this shaded region let me just separate this out as well

so just by comparing these areas you can see that the work done in case of the yellow path is much more than the work

done in case of the Green Path although we are getting from the same initial position to the same final position

which is why we know that work is a path function and right here you can also see how

pressure and volume are State functions because irrespective of the path we take the yellow one or the green one the

initial position is at 1 and the final position is at two so as long as this is the initial position and this is the

final position the difference in pressure or difference in volume is going to be the value at 2 minus the

value at 1 and unlike the work done it does not depend on the path we take which is why both of these are State

functions