Partial changes, infinitesimal

Because partial derivatives are used so prominently in thermodynamics (See Maxwell s Relationships), we briefly consider the properties of partial derivatives for systems having three variables x, y, and z, of which two are independent. In this case, z = z(x,y), where x and y are treated as independent variables. If one deals with infinitesimal changes in x and y, the corresponding changes in z are described by the partial derivatives ... 

The subscripts indicate the variables that are held constant when taking the partial derivatives. Likewise, one can chose to treat y and z as independent variables, thereby writing x in terms of these variables, such that X = x(y,z). In this case, we can obtain the following expression for infinitesimal changes in these independent variables ... 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

Ordinarily a dispersed mi can be represented as the product of density and partial specific volume (d, ) and v, having dimensions of area times length (Ax), any infinitesimal change in Ax (dA dx) is accompanied by an infinitesimal change in F (dF). Substituting in Eq. (4.64) and rearranging,... 

It is convenient at this stage to introduce a standard free energy change AGd for the reaction. This is the value of AGd when the reaction is advanced infinitesimally and reversibly under conditions where all gaseous constituents are at unit partial pressure (usually one atmosphere) at the temperature of interest. It may not be possible actually to execute the reaction of interest under such conditions, but this fact does not detract from the correctness of the assertion. The definition is also consistent with the fact that we had earlier... 

If the changes are finite rather than infinitesimal, but are small enough that the values of the partial derivatives are not appreciably affected by the changes, we have approximately... 

Because of their rigid cell walls, large hydrostatic pressures can exist in plant cells, whereas hydrostatic pressures in animal cells generally are relatively small. Hydrostatic pressures are involved in plant support and also are important for the movement of water and solutes in the xylem and in the phloem. The effect of pressure on the chemical potential of water is expressed by the term VWP (see Eq. 2.4), where Vw is the partial molal volume of water and P is the hydrostatic pressure in the aqueous solution in excess of the ambient atmospheric pressure. The density of water is about 1000 kg m-3 (1 g cm-3) therefore, when 1 mol or 18.0 x 10-3 kg of water is added to water, the volume increases by 18.0 x 10-6 m3. Using the definition ofV,., as a partial derivative (see Eq. 2.6), we need to add only an infinitesimally small amount of water (dnw) and then observe the infinitesimal change in volume of the system (dV). We thus find that Vw for pure water is 18.0 x 10-6 m3 mol-1 (18.0 cm3 mol-1). Although Vw can be influenced by the solutes present, it is generally close to 18.0 x 10-6 m3 mol-1 for a dilute solution, a value that we will use for calculations in this book. 

The heat of reaction (sometimes called the differential or partial heat of reaction to distinguish it from that defined previously) may then be defined as the heat evolved in this infinitesimal process per unit change of e, whence the relation dQ = dH and equation (40) show that the heat of reaction may be written as... 

In a homogeneous mixture each component i has a chemical potential pt, defined as the partial molar free energy of that component (i.e., the change in Gibbs energy per mole of component ti added, for addition of an infinitesimally small amount). It is given by... 

Each term of this equation is the change due to the change in one independent variable, and each partial derivative thus is taken with the other independent variables treated as constants. If we make the changes dV in V, dT in T, and dn in n these changes affect P separately, and we can write for the total infinitesimal change in... 

This equation delivers the value of dy corresponding to arbitrary infinitesimal changes in x and z, so it is still correct if we choose values of dz and dx such that dy vanishes. We now divide nonrigorously by dx, and interpret the quotients of differentials as partial derivatives, remembering that y is held fixed by our choice that dy vanishes. 

If this is the differential of a function, then M and N must be the appropriate derivatives of that function. Pfaffian forms exist in which M and N are not the appropriate partial derivatives of the same function. In this case du is called an inexact differential. It is an infinitesimal quantity that can be calculated from specified values of dx and dy, but it is not equal to the change in any function of x and y resulting from these changes. 

On the other hand, the number of possible states for adsorbed molecules, corresponding to different partial molar areas cOj, can be quite large. Theoretically one can assume a continuous change of co between cOmin and cOmax, and successive values varying from each other by an infinitesimal increment of the molar area Aco. The transition from a discrete to a continuous reorientation model can be performed formally, replacing the summation in Eqs. (2.78) and (2.89) by an integration. 

A dependent variable is a function of the independent variables. The total differential of a dependent variable is an expression for the infinitesinial change of the variable in terms of the infinitesimal changes of the independent variables. As explained in Sec. F.2 of Appendix F, the expression can be written as a sum of terms, one for each independent variable. Each term is the product of a partial derivative with respect to one of the independent variables and the infinitesimal change of that independent variable. For example, if the system has two independent variables, and we take these to be T and V, the expression for the total differential of the pressme is... 

Thus, we may interpret the partial molar volume of B as the volume change per amount of B added at eonstant T and p when B is mixed with sueh a large volume of mixture that the eomposition is not appreeiably affeeted. We may also interpret the partial molar volume as the volume ehange per amount when an infinitesimal amount is mixed with a finite volume of mixture. 

The partial molar enthalpy of a species is the enthalpy change per amount of the species added to an open system. To see why the particular combination of partial molar enthalpies on the right side of Eq. 11.2.6 is the rate at which enthalpy changes with advancement in the closed system, we can imagine the following process at constant T and p An infinitesimal amount dn of N2 is removed from an open system, three times this amount of H2 is removed from the same system, and twice this amount of NH3 is added to the system. The total enthalpy change in the open system is dH =... 

Thus, dU has one term that varies with temperature and one term that varies with volume. The two partial derivatives represent slopes in the plot of C7 versus T and V, and the total infinitesimal change in U, dU, can be written in terms of those slopes. Figure 2.8 illustrates a plot of U and the slopes that are represented by the partial derivatives. 

The term differential is sometimes added to enthalpy changes where infinitesimal (i.e., very small) amounts were added to a veiy large amount of either solution or pure component. These enthalpy changes are usually called partial (molar or specific) enthalpies of solution/mixing ... 

Z. is described as the partial molar property of component i in the solution and may be defined as the rate of change of the extensive property Z when an infinitesimal amount of component i is added to the system, keeping the temperature.pressure and amount of all other components constant. 

The first term in Eq. 2, U/Sf )y f dT, consists of the partial differential coefficient SU/sT)y fi which gives the change of U for a unit change in T if all other variables, V and n in the present case, are kept constant. This partial differential coefficient is the heat capacity as defined in Fig. 1.2. It must be multiplied with the actual, infinitesimal change in temperature, dT. 

Second, the balance is taken over an incremental space element, Ac, Ar, or AV. The mass balance equation is then divided by these quantities and the increments allowed to go to zero. This reduces e difference quotients to derivatives and the mass balance now applies to an infinitesimal point in space. We speak in this case of a "difference" or "differential" balance, or alternatively of a "microscopic" or "shell" balance. Such balances arise whenever a variable such as concentration undergoes changes in space. They occur in all systems that fall in the category of the device we termed a 1-D pipe (Figure 2.1b). When the system does not vary with time, i.e., is at steady state, we obtain an ODE. When variations with time do occur, the result is a partial differential equation (PDE) because we are now dealing with two independent variables. Finally, if we discard the simple 1-D pipe for a multidimensional model, the result is again a PDE. 

---