Internal energy molar description

Here, dUc represents the change of internal energy due to the addition of mass during the chemical process, and /r is the chemical potential that provides a resistance when one mole of substance is added to the system. Note that the chemical potential [x is an intensive variable and the amount of substance n is the extensive variable. Here a molar mass description is used. In most of the textbooks in chemistry and chemical physics, molar mass descriptions are employed however, in continuum mechanics we need to use a concentration-based notation. Note that the chemical potential ii corresponds to the Gibbs energy for concentration c = 1 (shown later). 

Note 3.10 (Molar description, concentration, units and dimension). Let us consider the relationship between the internal energy per unit mass Ua, the chemical potential per unit mass pia and the mass concentration c. The dimension of duajdt is [J/(MT)], and this dimension is the same as that for pia dca/dt ( a). Since the dimension of pia is [J/M], Ca is dimensionless. Therefore the concentration which is conjugate to the chemical potential iia is simply the mass concentration q, and the change of internal energy due to a chemical process can be written as d(Uc)ct = pia dCa ( ). ... 

Here 5 is the exchangeable entropy in the molar description. Note that the internal energy per unit mass u [J/M] (M is the unit of mass) is related to the internal energy per unit substance U [J/mol] by mu = U where m is the molecular weight. Most of the textbooks using the molar description employ the pressure p (compression positive) instead of stress a (cf. Sect. 3.2.4). We here use the pressure P and volume V (expansion positive) in the molar descriptions. 

Thermodynamic functions (Continuum mechanics) (Molar description) Internal energy u(e, s ) U(V,S )... 

In Sect. 3.2.4 the internal energy was treated as a sum of the mechanical and heat energies. Here, in addition, we discuss the effect of an electrostatic field (p on the change of internal energy d Ug under the molar description. 

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