Heat transfer at constant volume

FIGURE 6.12 A bomb calorimeter is used to measure heat transfers at constant volume. The sample in the central rigid container called the bomb is ignited electrically with a fuse wire. Once combustion has begun, energy released as heat spreads through the walls of the bomb into the water. The heat released is proportional to the temperature change of the entire assembly. 

Because heat transferred at constant volume can be identified with the change in internal energy, AU, we can combine Eq. 8 with C = q/AT and define the heat capacity at constant volume, Cv, as... 

Heat Transfer at Constant Volume Bomb Calorimeters... 

Heat transfer at constant volume AU = q + w = q. Reactions to measure AU are performed in a constant volume calorimeter called a bomb calorimeter. 

Heat transfer at constant volume, qy (notation not used in the text)... 

For a given chemical change, the heat transferred at constant pressure (qp) is equal to the heat transferred at constant volume (qy) only if there is no net consumption or production of gas (i.e., when AWgas = 0). 

If the definition of work is limited to mechanical work, an interesting simplification is possible. In this case, AE is merely the heat exchanged at constant volume. This is so because if the volume is constant, no mechanical work can be done on or by the system. Then AE = q. Thus AE is a very useful quantity in constant volume processes. However, chemical and especially biochemical processes and reactions are much more likely to be carried out at constant pressure. In constant pressure processes, AE is not necessarily equal to the heat transferred. For this reason, chemists and biochemists have defined a function that is especially suitable for constant pressure processes. It is called the enthalpy, H, and it is defined as... 

If the heat is transferred at constant volume and no non-expansion work is done,... 

If heat transfer takes place at constant volume, the magnitude is defined as heat capacity at constant volume (Cy) and is equivalent, as we have seen, to the partial derivative of the internal energy of the substance at constant volume and composition ... 

Cp = heat capacity at constant pressure C = heat capacity at constant volume gc = gravitational constant (numerical values in Table A1) h = individual heat transfer coefficient H = enthalpy k=thermal conductivity k=C iCv... 

Alternatively, a reaction might be carried out in an open flask or other apparatus that keeps the pressure constant and allows the volume of the system to change freely. In such a case, AU 0 and the energy change in the system might be due to both heat transfer and PV work. We indicate the heat transfer at constant pressure by the symbol qp ... 

M FIGURE 8.9 Diagram of a bomb calorimeter for measuring the heat evolved at constant volume (AE) in a combustion reaction. The reaction is carried out inside a steel bomb, and the heat evolved is transferred to the surrounding water, where the temperature rise is measured. 

Cy is the heat capacity at constant volume. We have used the partial derivative notation (see Appendix A) for dUr/dTr, the rate of change of internal energy with temperature when volume is held constant. We cannot use a partial derivative for 8qr/dTr because, as we discussed, 8qr is the amount of heat transferred and not the change in something, which is required for a derivative. The added heat for a finite temperature range may be found by integrating Eq. (16) in the form 8cy = Cy dTy. ... 

Most chemical reactions are done under constant (atmospheric) pressure conditions rather than at constant volume, so it is desirable to relate the heat transferred at constant pressure, q, to some state property analogous to the internal energy, U. 

Enthalpy (FT) Measured in joules, H = E + pV the amount of energy in a system capable of doing work the sum of the internal energy of the system plus the product of its volume times the pressure on it. The change in enthalpy (AH) is equal to the heat transferred at constant pressure. 

By contrast, heat exchanges are loosely defined as disorderly transfers of energy. Work comes in several flavors, whereas heat is heat whether the source is a Bunsen burner, hotplate, or acid solution mixed with base. The fine print is important just as in work exchanges. Thus, the heat received by a system at constant pressure is not the same as heat received at constant volume. 

Other Refrigeration Methods. Cryocoolers provide low temperature refrigeration on a smaller scale by a variety of thermodynamic cycles. The Stirling cycle foUows a path of isothermal compression, heat transfer to a regenerator matrix at constant volume, isothermal expansion with heat transfer from the external load at the refrigerator temperature, and finally heat transfer to the fluid from the regenerator at constant volume. 

Lindemann <8> has made an interesting application of the new theory in the determination of the frequency of atomic vibration, r, from the melting-point. He assumes that at the melting-point, T the atoms perform vibrations of such amplitude that they mutually collide, and then transfer kinetic energy like the molecules of a gas. The mean kinetic energy of the atom will then increase by RT when the liquid is unpolymerised and the fusion occurs at constant volume this is the molecular heat of fusion. 

Heat Transfer enough energy at constant volume to raise the temperature to its final value (317 K), and use AU = q. 

We have seen that a constant-pressure calorimeter and a constant-volume bomb calorimeter measure changes in different state functions at constant volume, the heat transfer is interpreted as A U at constant pressure, it is interpreted as AH. However, it is sometimes necessary to convert the measured value of AU into AH. For example, it is easy to measure the heat released by the combustion of glucose in a bomb calorimeter, but to use that information in assessing energy changes in metabolism, which take place at constant pressure, we need the enthalpy of reaction. 

Liquid is fed continuously to a stirred tank, which is heated by internal steam coils (Fig. 1.21). The tank operates at constant volume conditions. The system is therefore modelled by means of a dynamic heat balance equation, combined with an expression for the rate of heat transfer from the coils to the tank liquid. 

The other extreme case is the adiabatic change, which occurs with no heat transfer between the gas and the surroundings. For a reversible adiabatic change, k = y where y = Cp/Cv, the ratio of the specific heat capacities at constant pressure (Cp) and at constant volume (C ). For a reversible adiabatic change of an ideal gas, equation 6.27 becomes... 

The operation of the reciprocating internal combustion engines represents a compromise between the Otto and the Diesel cycles, and can be described as a dual combustion cycle. Heat transfer to the system may be considered to occur first at constant volume and then at constant pressure. Such a cycle is called a dual cycle. 

A , the change in energy, equals the heat transferred to or from a system at constant volume A =... 

Calculate the final temperature and the change in internal energy when 500 J of energy is transferred as heat to 0.900 mol 02 at 298 K and 1.00 atm (a) at constant volume (b) at constant pressure. Treat the gas as ideal. 

If the change in temperature is carried out at constant volume, we use the constant-volume heat capacity, Cv. If the change is carried out at constant pressure, we use the constant-pressure heat capacity, CP. This formula also applies when the transfer is reversible, so we can write... 

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