P(dV) Work-done In Thermodynamics/ Displacement Work

As we know that in thermodynamics, the system and its surroundings (or two different systems) interact with each other in the form of mass and energy transfer across the boundary. And "the transfer of energy between the system and the surrounding takes place in two ways, namely heat and work". Here we will study about thermodynamic work. 

First of all we should know, What is work??? And how can we define work in thermodynamic???
WORK-DONE
In physics (or in practice), "When a force is acting on an object and due to this force, the object is displaced from one place to another in the direction of the force (or may be in the opposite direction), then the product of force and displacement shows the amount of work done on the object".
Mathematically;
           Work-done ＝ force×displacement
            ➩ { Work-done, dW ＝ F×dz }
Where,
     F➛ is the amount of force (or a
         component of force acting along
           the line of displacement 'dz'.
By convention, the value of work can be positive, negative and zero depending on the displacement of an object.

Whereas in thermodynamics, work is a way of energy transfer between the system and the surrounding. It is defined as;
"Work is the transfer of mechanical energy between the system and the surrounding (or between two systems". 
Or
"Any displacement in the system's boundry, work is said to be done". This, the work is a boundary phenomena. This is observed when the energy crosses the boundary of the system.
         Work➝ Macroscopic Motions
➛ Work stimulates the organized motion of the particles in one direction only.
➛ Work is not a form of energy.
➛ Work is a high grade energy while heat is a low grade energy.
➛ Work transfer between the system and surrounding when the driving force between them is something othar than temperature difference (or any mechanical driving force).
➛ When there is no transfer of work between the system and the surroundings, then we can say that the system and the surroundings are in mechanical equilibrium with each other.

FORMULA FOR WORK
The work which accompanies a change in volume of a fluid is often encountered in thermodynamics. A common example of this work is the expansion or compression of a fluid confined in a cylinder resulting from the movement of piston. 
Work-done in thermodynamic is given by a formula when pressure is constant;  
            ➩ W ＝ - ( P ext × ∆V)
Where,
         Pext - external pressure
          ∆V - change in volume of fluid
When the pressure is variable then the formula for the thermodynamic work-done is given as; 
This work can be performed by the system on the surroundings or by the surrounding on the system. Depending on the sign convention, this work can be positive and negative.
"This work is also known as the "pdv" work in thermodynamics and is always defined for a closed system".

Derivation Of "PdV" Work Formula
Assume that a gas (or fluid) is confined in a cylinder over which a movable piston is attached. External pressure exerted on the piston by atmosphere air. [P(gas) > P(ext)]
Since, external pressure is less than the pressure exerted by the gas which is confined in the cylinder therefore the piston moves '∆x' distance in upward direction and the volume of the gas increases in this situation. Thus, the gas is expanded in this case and due to that the gas perform some work on the surrounding and due to that, the energy of the gas confined in a cylinder is decreased.
The work-done in this case is given as;
➩ W ＝ force × displacement
➩ W ＝ - F×∆x 
Let area of the piston is A and piston moves by infinitesimal distance ∆x due to the pressure force of the gas acting over the piston. 
Pressure force exerted by gas over the piston will be calculated as, F= P. A
Thus,
➩ W＝ - P(gas) × A× ∆x
 ( ∵ area×height ＝ volume
                       ∴ A×∆x＝ ∆V )
➩ { W ＝ - P(gas) × ∆V }
Work-done for expansion process is negative because the change in volume of gas confined in a cylinder is positive whereas work-done for compression process is positive because the change in volume of gas confined in a cylinder is negative. [ Since, P(gas)<P(ext) ]
Thus, ➩ { W ＝ P(ext) × ∆V }
Expansion➝ work-done negative
Compression➝ work-done positive
☛ The above equation of work-done is valid where the pressure is constant. If pressure is not constant and it varies with volume (that is, when pressure is defined as a function of volume) then we use another equation to calculate the work done. This equation is given as;
☛ The expansion/compression work on the PV-diagram where we can not easily find the pressure as a function of volume, the magnitude of the work-done by gas over the piston will be  calculated by determining the area under the PV-curve.
Thus, for a closed system the work-done is calculated by calculating the area under the Pv-curve.

Sign Convention
Let us see here the sign conventions used for work energy transfer....
(1) (Expansion work)
If work is done by the system over its surrounding, quantity of work energy transferred will be considered as positive.
     (Compression work)
If work is done over the system from its surrounding, quantity of work energy transferred will be considered as positive.

PdV Work For different processes
For An Isobaric Process,
                      ➩ dP= 0
               ➩ [ W＝ P{V₂–V₁} ]
For An Isochoric Process, 
                      ➩ dV= 0
               ➩ [ W＝ PdV ＝ 0 ]
For An Isothermal Process, 
                      ➩ dT= 0
           ➩ [ W＝ P₁V₁ {(ln V₂) – ( ln V₁)}]
                                     Or
           ➩ [ W＝ P₁V₁ {(ln P₁) – ( ln P₂)}]
For Polytropic Process, 
                     ➩ PVⁿ =constant)
            ➩ [ W＝ (P₁V₁–P₂V₂) /(n–1) ]


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