Energy partial molar internal

Here, M, may represent the partial molar internal energy Uit the partial molar enthalpy H(, the partial molar entropy Sif the partial molar Gibbs energy Gt, etc. Comparison of Eq. (10.1) with Eq. (11.2) written for the Gibbs energy shows that the chemical potential and the partial molar Gibbs energy are identical, that... 

Thus, for the fth component in a system, partial molar internal energy=(dEV ni)jpny Tri = partial molar enthalpy = pH/ ni)rPnin3 = H partial molar entropy = (dS/ dn ... 

Tliis equation defines the partial molar property of species i in solution, where the generic symbol Mt may standfor the partial molar internal energy t/, the partial molar enthalpy //, the partial molar entropy 5,, the partial molar Gibbs energy G,, etc. It is a response function, representing the change of total prope ity n M due to additionat constant T and f of a differential amount of species i to a finite amount of solution. 

Figure 5.5 Plots of (a) the partial molar internal energy C Li arid (b) entropy Su of lithium ions at constant volume V and temperature T with respect to lithium content (1 —8), theoretically calculated from the mean-field approximation (A) and the Monte Carlo simulation (0) at T = 298 K. The partial molar enthalpy and...

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

The partial molar internal energy of transferring a mole of component 2 from its pure liquid to the solution can be written as... 

Equation 9.1-4 indicates that the partial molar internal energy of species i in an ideal gas mixture t a given temperature is equal to the pure component molar,internal energy of that component as an ideal gas at the same temperature. Similarly, Eq. 9.1-5 establishes that the partial molar volume of species i in an ideal gas mixture at a given temperature and pressure is identical to the molar volume of the pure component as an ideal gas at that temperature arid pressure. 

Here uG — is the difference in the partial molar internal energy of the gas G and the adsorbed layer 1, and —p (iV/tym)s is the adiabatic differential heat of compression. Reversible adsorption can only occur at temperatures such that diffusion of the adsorbed gas is fast enough to establish an equilibrium distribution at the surface during the time of the measurements. This criterion holds for chemisorption as well, but implies that observations are made at correspondingly higher temperatures. 

The coefficient of dNi/dt in the summation of (6-4) is defined as the partial molar internal energy of species i, because differentiation with respect to mole numbers of component i is performed at constant T, p and mole numbers of all other species in the mixture. The unsteady-state mass balance for species i in a batch reactor,... 

When the product of v, and the partial molar internal energy of species i is summed over all components in the system, one obtains an exact expression for the molar internal energy change for the reaction (see Tester and Modell, 1997, pp. 769-770). In other words. 

To obtain the partial molar internal energy and enthalpy, we use the generic expression (4.1.22) to obtain... 

Partial molar properties obey the same equations as regular properties. For example, the partial molar internal energy, enthalpy, and volume are related through the equation,... 

From this result it also follows that the internal energy of mixing is zero, and the partial molar internal energy is equal to the internal energy of the pure component ... 

Therefore, the chemical potential is the partial molar Gibbs energy as they share the same independent variables T and P. It is not the partial molar internal energy, Helmholtz free energy, and enthalpy. 

The isosteric heat is also the heat which would be transferred to the surroundings in the transfer of a differential quantity of sorbate from the vapor phase to the adsorbed phase under isothermal and isobaric conditions. In a similar differential calorimetric measurement carried out at constant volume, the heat released to the heat bath would be H,- U,. This quantity is sometimes referred to as the differential heat of sorption, but the term is redundant and it is probably better to consider this quantity simply as the difference in partial molar internal energy. 

The change in internal energy is —Uidrii, and the change in volume is - where Ui and Vj are the partial molar internal energy and partial... 

Thus, in an ideal gas mixture the partial molar internal energy and the partial molar heat capacity at constant pressure, like the partial molar enthalpy, are functions only of T. 

U, Vi internal energy, partial molar internal energy of i... 

For example, the internal energy of ice and water in equilibrium is the sum of the partial molar internal energy of ice times the number of moles of ice plus the partial molar internal energy of liquid water times the number of moles of liquid. 

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