First law of thermodynamics and Cyclic process

Cyclic process:-
If a system performs the number of processes so that it comes back again to the initial state then the system the number of processes form a closed loop in a thermodynamic diagram which is called a thermodynamic cycle.

cyclic process

                                            

Sign convention of work and heat flow:-

sign convention for heat and work

                   

The first law of thermodynamics:-

The First Law of Thermodynamics states that energy can be converted from one form to another with the interaction of heat, work and internal energy, but it cannot be created nor destroyed, under any circumstances. Mathematically, this is represented as

ΔU=q+w

where

ΔU  is the total change in internal energy of a system,

q  is the heat exchanged between a system and its surroundings, and

w  is the work done by or on the system.

The algebraic sum of net heat and work interaction between a system and its surrounding in a thermodynamic cycle is zero.

                                                          ∑Q = ∑W
                           cycle        cycle
                                           ∮(Q-W)=0
                             ∮𝛅Q-𝛅W=0
►Closed cyclic integral of any point function is a zero.

             X is any point function 
                    ∮dx=0
        ∴𝛅Q-𝛅W=dx
             Q-W=Δx
   This is defined as internal energy.
  ∴𝛅Q-𝛅W=dE
  ⇒𝛅Q=𝛅W+dE
  ∴Q-W=ΔE
  ⇒Q=W+ΔE
    Q1-2-W1-2=E2-E1
    Q1-2=W1-2+E2-E1
E=u+K.E+P.E+Any other kind of energy
dE=du+d(K.E)+d(P.E)+d(..........)

 Internal energy is a point function.

Limitation of the 1st law of thermodynamics

Internal Energy:-
Internal energy is a property of system whose change in a process executed by the system equals to the difference between heat and work interactions by the system with its surroundings.

. In a thermodynamic process
       𝛅Q-𝛅W=dE
   where E is the internal energy
. The Internal energy comprises inter-molecular energy kinetic energy     potential energy of a system.

✸ For a closed or stationary system, the inter-molecular energy is the only component of internal energy and is usually denoted by the symbol "U".
                     Q1-2=W1-2+E2-E1
                     𝛅Q=𝛅W+dE
              ∮𝛅Q=∮dE+∮𝛅W

             E=u+K.E+P.E  (for stationary system K.E=0)
                    dE=du+d(K.E)+d(P.E) (small change in P.E is zero)
                    dE=du

             Q1-2=u2-u1+W1-2
                   𝛅Q=du+𝛅w
If we consider a closed system which perform only reversible displacement work that means Quasi static displacement work which is nothing but pdv work.
                   𝛅Q=du+𝛅w
             (for closed system 𝛅w=pdv)  
            𝛅Q=du+pdv

✸ The first law for a closed system in an infinitesimal process can be           written as
                 𝛅Q=du+𝛅w
✸ For a finite process between two states executed by a closed                  system.
                Q1-2=u2-u1+W1-2
✸ For a finite process for per unit mass

               
✸ For infinite
                 𝛅Q/𝛅m=du+pdv