Understanding Vertical Displacement and Time Taken in Thermodynamics
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Introduction
In this article, we delve into the concepts surrounding vertical displacement and time taken in the context of thermodynamics. Picture a scenario where you and a friend are in front of a building, attempting to travel from the ground floor (Point A) to the top floor (Point B). Each person chooses a different path: one takes the lift, while the other opts for the stairs. While both will reach the same final destination, the journey varies significantly in terms of time, offering an excellent analogy to explain state functions and path functions commonly used in thermodynamics.
Understanding Vertical Displacement
Definition of Vertical Displacement
Vertical displacement is defined as the direct distance between two points, irrespective of the path taken. In mathematical terms, if you start at Point A and end at Point B, the vertical displacement can be calculated as:
 Vertical Displacement (Δy) = y<sub>B</sub>  y<sub>A</sub>
Since both you and your friend start at the same initial position and end at the same final position, the vertical displacement from A to B remains the same for both travelers, regardless of the method used to reach that point.
The Importance of Initial and Final States
In practical terms, vertical displacement serves as a reminder of how we define positions in any physical journey. It highlights that only the initial and final states matter when considering displacement.
Time Taken: The Path Matters
Differences in Time Taken
In contrast to vertical displacement, the time taken to ascend to the top floor will differ significantly between the two paths. Assuming the lift operates efficiently, the time spent going up by the lift will generally be less than that taken using the stairs.
 Time Taken by Lift < Time Taken by Stairs
This scenario illustrates that while displacement is concerned only with the endpoints, the time taken (a crucial aspect) is dependent on the path taken. The distinctiveness between the variables becomes clearer, emphasizing how these two attributes interact differently based on the path chosen.
Thermodynamics: State Functions vs. Path Functions
As we shift our focus to thermodynamics, the link to vertical displacement and time taken can be drawn through the concepts of state functions and path functions.
Thermodynamic Variables
To explain a thermodynamic system transitioning between two states (State 1 to State 2), we employ several thermodynamic variables such as:
 Pressure (P)
 Temperature (T)
 Volume (V)
 Number of moles (N)
Each of these variables can describe the initial state (P<sub>1</sub>, T<sub>1</sub>, V<sub>1</sub>, N<sub>1</sub>) and the final state (P<sub>2</sub>, T<sub>2</sub>, V<sub>2</sub>, N<sub>2</sub>) of the system. These state variables can be used to depict changes similar to how we measured vertical displacement.
State Functions
State functions depend solely on the initial and final states of the system, irrespective of the process used to get there. Some common examples include:
 Change in Pressure (ΔP)
 ΔP = P<sub>2</sub>  P<sub>1</sub>
 Internal Energy (U)
 Represents the total energy of a system, incorporating kinetic and potential energy of molecules.
 Entropy (S)
 A measure of disorder or randomness in a system.
These functions signify that once the end points are known, the specific details of the path taken are irrelevant.
Path Functions
In contrast to state functions, path functions, which include work (W) and heat (Q), depend on the pathway taken between the states:
 Work Done (W): The work done by or on the system varies based on the process path.
 Heat Transfer (Q): The heat added or removed also depends on the pathway taken.
Example: PressureVolume Diagram
To illustrate the contrast between path functions and state functions, consider a PressureVolume (PV) diagram:

Two paths from (P<sub>1</sub>, V<sub>1</sub>) to (P<sub>2</sub>, V<sub>2</sub>):
 First Path (Yellow): Increasing pressure first while keeping volume constant followed by a change in volume.
 Second Path (Green): Changing volume first while keeping pressure constant followed by a change in pressure.

These two distinct paths will yield different amounts of work done, demonstrable as the area under the curve in the PV diagram.
 Path A: More area under the curve, indicating more work done.
 Path B: Less area under the curve, indicating less work done.
Conclusion
In summary, the examination of vertical displacement and time onset in a physical scenario expands into the realm of thermodynamics through state functions and path functions. While displacement between two points relies solely on initial and final positions, time taken to cover that distance is pathdependent. Similarly, state properties like internal energy, entropy, and pressure differ from pathdependent quantities such as work and heat, underscoring the complexities and the different concepts within the field of 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
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 catchall 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
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