The following literature may be useful for your studies

There is no difference of quality between the different forms of energy. However, observations do show that heat and work are not equal: one can always transform work to heat, but the reverse is much more complicated and often not possible. For instance, we have learned in the second chapter that in order to do mechanical work, e.g., a gas-system has to change its volume against an external pressure. If this external pressure is zero and an ideal gas expands (certainly irreversibly) no work would be done and no heat would be absorbed. That's puzzling because the internal energy doesn't change; there is no energetic "motivation" for the gas to expand. Nevertheless, the gas will evenly fill the larger volume that became available. What's the "driving force" for that change? Clearly, there must be some property, called entropy that governs the transformation of systems. 

Ref: casey.brown.edu/research/crp/Edu/.../Chem201_3_second_law.htm

Dear Dr. Toleda;   I am sorry to say that you guys are trying to rediscover classical thermodynamics by going back to Carnot (1824), Joule (1840-45), Helmholtz (1847), and Clausius  (1850). To day we are adopting the formulation of  Max Born (1921), which is surpassed in rigour and clarity all earlier attempts.   Prof. Dr.  Krishna is on the right step by spelling the secret word of  'Entropy'  or ' the Second Law of Thermodynamics after by making a  very successful 'gedankeexperiment'. Best Regards

Heat is a form of energy in transition states between two or more bodies that are at different temperatures. The quality of this form of energy as its exergy (are related concepts) depends on the value of the difference in temperatures. In the case of isothermal process heat is transmitted to the same temperature (hot and cold sources) and does not imply that thermodynamic irrreversibility ideally this case being a well explained in the theory of heat engines Carnot limit.

Okay!  Then please try to answer the following questions using Carnot Concept.  What is Temperature? What is energy? What is work? What is Heat?  How heat, energy and work are connected analytically?  What is the mathematical definition of entropy?  What type of thermodynamic quantity is changing  during the infinitesimal  isothermal process under constant pressure or volume?  Thermodynamics is a mathematical science like algebra or group theory, which relies on the few hypothesis, and  the rest is the logical manipulations (calculus). (Einstein).

The classical engineering thermodynamics  as You and I, we learned in mechanical engineering departments  can not take us too far. I was trained as a regular Navy Officer, and graduated first in my class. But I learned real chemical thermodynamics (including the irreversible one) by taking special courses from the Chemistry Department at Stanford while I was majoring  Material Science and Solid State Physics (minor) in the years of 1960-1965.   Best Regards 

Lest us  take the case of solidification of an alloy at constant temperature and pressure:

WE define a new thermodynamic function 'enthapy'  as  H=U+ PV .  Then write the first law as  dU = T dS-PdV and combine this with definition of enthapy. You get: dH=TdS +VdP.  For an isobaric process  dH=TdS

This formula can be used to calculate the enthapy change during the isothermal and isobaric solidification process by simple integration operation, which gives:  DH=TDS = DQ, where Q is heat by definition (D  finite difference operator) which is for  one mole quantity  of alloy known as the Heat of Fusion or Enthapy of Fusion under the constant T and P. Similarly,  one can define another function known as Gibbs Free energy by  G=U-TS+ PV, and one shows by calculus that dG = -SdT + VdP,which becomes zero for the isothermal and isobaric process under the equilibrium.  dG=0  or  DGT,P=0. This is very imported relationship in most of the chemical  reactions, which takes place in atmospheric conditions. One can also show using the definition of entropy by Max Born (1921) that for irreversible (natural) processes  dGT.P EQL  =0

One may do the same approach for the isothermal and isovolumic processes by defining Helmholzt Free energy F=U-TS, and using the differential form such as:  dF= -SdT-PdV  to show  DFT,V=0, and   dU=TdS  by integration for one mole of alloy:  DU=TDS = DQ  Heat of Fusion under the isothermal and isovolumic condition: QED  Similarly  dFT.V  EQL 0. 

It is the "net" heat, in other words, the exergy of the heat, becomes work.

Dear Ying you are in right track. It is the  transformation of the form of energy from one to another keeping its entity invariant.

Thanks Tarik.

True in a reversible isothermal process you can transform a given quantity of heat into an equivalent quantity of work.

But the crucial point is that, you must do that cyclically (repeatedly). If you do an isothermal cycle, you will notice that you will not be able to convert heat into work, because at the end of the cycle the heat reservoir suffers no change and we don't get any work output. Therefore, the isothermal process doesn't acknowledge the difference in quality of heat and work (or other forms of energy). Besides, we can convert a given amount of work into an equivalent quantity of heat in a cyclic process. Therefore, such a process also will not allow us to understand the fact that the quality of heat and work are really different!

It is therefore necessary for us to perform a cyclic process where heat is the input and work (or any other form of energy) is the output. When we do this we will find that we don't get an equivalent quantity of work (or any other form of energy) output for a given quantity of heat input. This is the reason that leads to the conclusion that the quality of heat and work (or any other form of energy)  are different.

With the development of   special theory of relativity by  Einstein (1905) and later put into geometrical format in terms of union of space and time by his mentor Hermann Minkowski (1907),  we enlarged the law of conservation energy to include  MASS into the scenario.

When a closed system undergoes a process, no mass changes occur. Similarly when we consider a cyclic process no mass changes occur.

Therefore, inclusion of mass changes in the 'enlarged' law conservation of energy, need not bother us, for the processes we are considering - conversion of heat into work. 

Dear Roberto I am sorry to say that  it is nonsense to talk about the quality of heat  if one wishes to stay in the domain of thermodynamics.  There are many forms of  energy the heat  and mass are  few of  them. What is important  that is the universal law of conservation of energy!!

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity