Total differential of the volume

To apply this criterion of exactness to a simple example, let us assume that we know only the expression for the total differential of the volume of an ideal gas [Equation (2.6)] and do not know whether this differential is exact. Applying... 

Consider the volume as an example of an extensive variable. The total differential of the volume (T, P constants) is... 

The total differential of the internal energy U of a system can be written as a function of independent state variables such as the temperature, volume and composition of the system as shown in Eq. 5.1 ... 

There are several examples where we can relate thermodynamically measurable parameters to the molecular properties. The link between the internal pressure and the van der Waals constants, a and b is a good example. The internal pressure of a fluid is defined from pure thermodynamics if we consider the total differential of the internal energy of a fluid as a function of its entropy and volume, U = f(S,V)... 

The resemblance between Eq. (101) and the total differential of the internal energy with respect to the number of moles of a chemical species, n, and the volume of the container, V, at absolute zero ... 

We have again to divide the system into the phases a and i.e. define the volumes Va and V. Adopting the device of a dividing surface, it turns out that the total differential of the surface free energy is... 

Here we have divided by i and introduced the molar quantities for entropy and volume. Obviously in Eq. (7.16) no term that gives information on the size of the system is seen. Moreover, the chemical potential is expressed as a total differential of the independent variables T, p. Thus there should be a function of the form... 

For a volume-explicit equation of state, an analogous expression can be derived, starting from Eq. (C.26). In this case, the term (8P/3T) must be replaced. Setting up the total differential of the specific volume... 

The presence of the membrane makes this system different from the multiphase, multicomponent system of Sec. 9.2.7, used there to derive conditions for transfer equilibrium. By a modification of that procedure, we ean derive the conditions of equilibrium for the present system. We take phase P as the reference phase because it includes both solvent and solute. In order to prevent expansion work in the isolated system, both pistons shown in the figure must be fixed in stationary positions. This keeps the volume of each phase constant dF = dF = 0. Equation 9.2.41 on page 236, expressing the total differential of the entropy in an isolated multiphase, multicomponent system, becomes... 

For a second-order transition, we will show the development of two similar relationships, one which results from the volume continuity at the transition and the other from the entropy continuity. In the first instance, we can write that the total differential for the volume in the liquid 1 and the glass g are the same. 

By differentiating Eq.(6) with respect to time, considering that the variations of the volume and total pressure are negligible, and using the enthalpy definition. Hi = CpiT, where Cp is the heat capacity of i-reactant (kJ/mol °C), Eq.(5) can be written as follows ... 

Conversion Formulas. Often no convenient experimental method exists for evaluating a derivative needed for the numerical solution of a problem. In this case we must convert the partial derivative to relate it to other quantities that ate readily available. The key to obtaining an expression for a particular partial derivative is to start with the total derivative for the dependent variable and to realize that a derivative can be obtained as the ratio of two differentials [8]. For example, let us convert the derivatives of the volume function discussed in the preceding section. 

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

It is of interest to obtain thermodynamic relations that pertain to the dielectric medium alone. The system is identical to that described in Section 14.11. However, in developing the equations we exclude the electric moment of the condenser in empty space. We are concerned, then, with the work done on the system in polarizing the medium. Instead of D we use (D — e0E), which is equal to the polarization per unit volume of the medium, p. Finally, we define P, the total polarization, to be equal to Fcp. Now the equation for the differential of the energy is... 

When we assume that the mole numbers of the materials that compose the solenoid are constant, the energy of the total system is a function of the entropy, volume, mole numbers of the material within the solenoid, and the magnetic induction. Thus, we have for the differential of the energy of the system... 

An extensive variable may be converted into an intensive variable by expressing it per one mole of a substance, namely, by partially differentiating it with respect to the number of moles of a substance in the system. This partial differential is called in chemical thermodynamics the partial molar quantity. For instance, the volume vi for one mole of a substance i in a homogeneous mixture is given by the derivative (partial differential) of the total volume V with respect to the number of moles of substance i as shown in Eq. 1.3 ... 

Write transient balances for the total mass of the tank contents and the mass of A in the tank. Convert the equations to differential equations for V i) (the volume of the tank contents) and Ca(0 (the concentration of A in the tank) that have the form of Equations 11.4-1 and 11.4-2, and provide initial conditions. 

The total rate of particle growth is expressed by Eq. (39), ouly the micellar volume t>m should be exchanged by the volume of the precipitated oligomers. We do not differentiate between the volumes of dead and living particles, since the particles rapidly change from active to inactive and vice versa. 

In Eq. (3) the initial reactivity is given by the parameter A . In Eqs. (3-5) the label o refers to the initial charcoal structure which is characterised by the reaction surface area per unit volume, S, the total length of the pore per unit solid volume, Lq, the particle radius, Rg, and the porosity, Cg. The surface reaction is characterised by the reaction rate constant K, and the reaction order n with respect to the reactant gas concentration C, Differentiating Eq. (2) with respect to t for o oo (i.e., the reaction on the outer particle surface is neglected) one obtains... 

U, H, and S as Functions of T and P or T and V At constant composition, molar thermodynamic properties can be considered functions of T and P (postulate 5). Alternatively, because V is related to T and P through an equation of state, V can serve rather than P as the second independent variable. The useful equations for the total differentials of U, H, and S that result are given in Table 4-1 by Eqs. (4-22) through (4-25). The obvious next step is substitution for the partial differential coefficients in favor of measurable quantities. This purpose is served by definition of two heat capacities, one at constant pressure and the other at constant volume ... 

Two-thirds of total body water is distributed intracellularly while one-third is contained in the extracellular space. Sodium and its accompanying anions, chloride and bicarbonate, comprise more than 90% of the total osmolality of the extracellular fluid (ECF), while intracellular osmolality is primarily dependent on the concentration of potassium and its accompanying anions (mostly organic and inorganic phosphates). The differential concentrations of sodium and potassium in the intra- and extracellular fluid is maintained by the Na+-K+-ATPase pump. Most cell membranes are freely permeable to water, and thus the osmolality of intra- and extracellular body fluids is the same. Symptoms in patients with hypo- and hypernatremia are primarily related to alterations in cell volume. It is therefore essential to understand the factors that cause changes in cell volume. 

Since the well-known familiar forms of general principles have been deduced and always written for a system, consider first a system coinciding with the control volume at the final state while including in the initial state the piston-cylinder assembly as well as the control volume. Let E, E2 and E, E"v denote the initial and final values of the total energy of the system and the control volume, respectively. The first law of thermodynamics for the system undergoing a differential process is2... 

To calculate the pore pressure response due to a volume source we use the Green s function based on the effective Biot theory. We write the coupled system of equations directly from the constitutive relations given by Biot (1962). These are the total stress of the isotropic porous medium, the stress in the porous fluid, the momentum balance equation for total stress, and the generalized Darcy s law. Following Parra (1991) and Boutin et al. (1987), the coupled system of differential equations in the... 

Differentiation and separation only occurs over a certain range of molecular sizes, typically between molecular weights of 2 kDa and 200 kDa, although this can be increased up to 1,000 kDa by the use of more specialised gels. This size range is dependent on the sizes of the pores and pore size distribution in the gel matrix. Retention volumes Vr are often used in size exclusion chromatography instead of retention times tR. The total volume V, of the separation column is the sum of the volume of the gel particles Vg, the volume of the solvent inside the pores, also called the intrinsical volume V and the volume of the free solvent outside the pores, the inter particle volume Vq ... 

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