Internal energy, changes at constant

Relating the enthalpy change to the internal energy change at constant pressure (185) ... 

We said before that internal energy might be thought of as hotness plus chemical energy. However, there can also be internal-energy changes at constant temperature. Suppose we have some mass of some substance in an absolutely rigid vessel. Now we transfer heat into the vessel. For this process Eq. 4.18 yields... 

AJJ in equation (7.15) is, strictly speaking, the internal energy change at constant pressure. Setting AH = in this equation is an approximation but one that is usually valid. 

We need to use three definitions the infinitesimal versions of the free energy change at constant temperature (dG = dH — T dS), the enthalpy change at constant pressure (dH = dU + PdV), and the internal energy change (dU = dw + dq). We substitute the second expression into the first and obtain... 

AP is the energy change at constant volume and temperature Tj, in the chemical reactions to produce the products which are the equilibrium composition at F25 2 - 2, i change in internal... 

A bomb calorimeter measures energy change at constant volume. A bomb calorimeter tells us the internal energy change in a reaction. (Recall that at constant volume q - ALL) In a bomb calorimeter, a steel container full of reactants is placed mside another rigid, thermally insulated container. 

Since so many chemical processes are conducted under constant pressure (for example in open beakers onto which the atmosphere exerts a constant pressure) rather than under constant volume conditions (in which systems need to be confined inside a closed vessel) we encounter changes in enthalpy, AH, more frequently than we do changes in internal energy, AU. At constant pressure, AP = 0 ... 

Once Ceai has been determined, the calorimeter can be used to measure the heat of combustion of other substances. Because a reaction in a bomb calorimeter occurs under constant-volume rather than constant-pressure conditions, the measured heat change corresponds to the internal energy change (At/) rather than to the enthalpy change (A//) (see Equations 5.6 and 5.11). It is possible to... 

Grady and Asay [49] estimate the actual local heating that may occur in shocked 6061-T6 Al. In the work of Hayes and Grady [50], slip planes are assumed to be separated by the characteristic distance d. Plastic deformation in the shock front is assumed to dissipate heat (per unit area) at a constant rate S.QdJt, where AQ is the dissipative component of internal energy change and is the shock risetime. The local slip-band temperature behind the shock front, 7), is obtained as a solution to the heat conduction equation with y as the thermal diffusivity... 

For a constant-volume system, an infinitesimal change in temperature gives an infinitesimal change in internal energy and the constant of proportionality is the heat capacity at constant volume... 

It is said that every substance has an internal energy (designated as E), and that the heat effect associated with a change at a constant volume and temperature is AE. As the molecules go from "state 1" to "state 2," AE = E2 - E,. This effect is exactly analogous to the heat effect that is associated with a change at constant pressure and temperature AH = H2 - Hx. The variables// and E are related by the potential of the system to expand or contract—that is, to the potential to be affected by PV work — by the explicit function... 

Suppose that a change in the system takes place at constant pressure and that during the change the internal energy changes by AU and the volume changes by AV. It then follows from the definition of enthalpy in Eq. 16 that the change in enthalpy is... 

What is the difference between the internal energy change AE and the enthalpy change AH Which of the two is measured at constant pressure, and which at constant volume ... 

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell-Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity K(fhl, ij) may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium. 

The application of Eqs. (2.4) and (2.5) is restricted to nonflow (constant mass) processes in which only internal-energy changes occur. Far more important industrially are processes which involve the steady-state flow of a fluid through equipment. For such processes the more general first-law expression [Eq. (2.3)] must be used. However, it may be put in more convenient form. The term steady state implies that conditions at all points in the apparatus are constant with time. For this to be the case, all rates must be constant, and there must be no accumulation of material or energy within the apparatus over the period of time considered. Moreover, the total mass flow rate must be the same at all points along the path of flow of the fluid. 

It has been observed experimentally that internal energy is nearly independent of pressure for solids and liquids at a fixed temperature, as is specific volume. Therefore, if che pressure of a solid or liquid changes at constant temperature, you mav write A(7 = 0 and AW [ = U A(P1/)] - V P. 

A temperature change AT from Tj leads to a change AO in specific internal energy. As AT - 0, the ratio AW/AT approaches a limiting value (i.e., the slope of the curve at Ti), which is by definition the heat capacity at constant volume of the substance, denoted by Cv... 

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