Legendre transforms internal energy equation

Legendre transformation does not affect the essential nature of a function and all of the different potentials defined above still describe the internal energy, not only in terms of different independent variables, but also on the basis of different zero levels. In terms of Euler s equation (2) the internal energy consists of three components... 

The successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

Physically, da i he change in the internal energy of the system with respect to an increase in the number of moles of species a, while holding all number of moles of all other species constant. The other fonns of the fiindamental equation of thermodynamics can be similarly generalized by performing the Legendre transform ... 

Step 2. Use the total differential of specific enthalpy in terms of its natural variables, via Legendre transformation of the internal energy from classical thermodynamics, to re-express the pressure gradient in the momentum balance in terms of enthalpy, entropy, and mass fractions. Then, write the equation of change for kinetic energy in terms of specific enthalpy and entropy. 

This is the product rule for the divergence of the product of scalar and a vector. The thermodynamic relation between specific enthalpy and specific internal energy via Legendre transformation i%h = u + p/p (see equation 29-20). Hence, the second term on the right side of (25-25) and the second and sixth terms on the right side of (25-26) can be combined as follows ... 

---