Capacity, heat

The heat capacity or specific heat of a unit mass of material is the amount of energy required to raise its temperature 1°C It can be measured either at constant pressure or constant volume. At constant pressure it can be larger than at constant volume, because additional energy is required to bring about a volume change against external pressure. 

The specific heat of amorphous plastics increases with temperature in an approximately linear fashion below and above Tg, but a steplike change occurs near the Tg. No such stepping occurs with crystalline types. 

For plastics, heat capacity is usually reported during constant pressure heating. Plastics differ from traditional engineering materials because their specific heat is temperature sensitive. 

The heat capacity is the amount of energy required to increase the temperature of a unit mass of material. It is commonly measured using a differential scanning calorimeter (DSC). The heat capacity depends on the resin type, additives such as fillers and blowing agents, degree of crystallinity, and temperature. A temperature scan for the resin will reveal the Tg for amorphous resins and the peak melting temperature and heat of fusion for semicrystalline resins. The heat capacities for LDPE and PS resins are shown in Fig. 4.15. 

The specific energy is the amount of energy required to increase a unit mass of material from ambient conditions to the process temperature, ft is easily calculated from the heat capacity by integrating as follows  

For most processing equipment, the low thermal conductivity of polymers strongly influences the overall heat transfer coefficient between the bulk of the polymer and the contacting metal surfaces, creating limitations in heat transfer rates. Heat transfer rates between processing equipment and the polymer depend on many factors, including thermal conductivity, machine clearances, and screw 

The temperature of molten polymer process streams is commonly measured using a thermocouple positioned through a transfer line wall and partially immersed in the polymer stream. Process stream temperature measurements that use an exposed-tip thermocouple, however, can be misleading since the temperature of the thermocouple junction is a balance between the heat transferred from the polymer stream and from the thermocouple assembly [39]. Due to the low heat transfer rate between the polymer and the exposed tip and the high thermal conductivity of the thermocouple sheath, the temperatures measured can be different by up to 35°C depending on conditions. Extrudate temperatures, however, can be accurately measured using a preheated, handheld thermocouple probe. This method minimizes thermal conduction through the probe sheath. 

Single-screw extruders and most other processing machines are volumetric metering devices. That is, the extruder will discharge a volume of resin for each revolution of the screw. Since the processor requires rate data in mass units (kg/h), the melt density is a needed physical property. The melt density for polymers is always less than the solid density of the material, and the melt density decreases further 

The heat capacity of a material is the heat required to raise the temperature of 1 unit weight of a substance 1 degree, and the ratio of the heat capacity of one substance to the heat capacity of water at 15°C (60°F) in the specific heat. 

The heat capacity of coal can be measured by standard calorimetric methods that have been developed for other materials (e.g., ASTM C-351). The units for heat capacity are Btu per pound per degree Fahrenheit (Btu/lb-0F) or calories per gram per degree Celsius (cal/g °C), but the specific heat is the ratio of two heat capacities and is therefore dimensionless. 

The heat capacity of water is 1.0 Btu/lb-°F (= 4.2 x 103 J/kg K), and thus the heat capacity of any material will always be numerically equal to the specific heat. Consequently, there has been the tendency to use the terms heat capacity and specific heat almost equivocally. 

Source Moisture Volatile Matter Fixed Carbon Ash 28-65°C 25-130°C 25-177°C 25-227°C 

FIGURE 7.1 Variation of specific heat with moisture content. (From Baughman, 1978, p. 172.) 

The heat capacity of a sohd quantihes the relationship between the temperature of a body and the energy supplied to it. The ideas date from times when heat was thought to be a fluid that could be transferred from one object to another. If a large amount of heat (energy) supplied to a body produced only a small temperature increase, the sohd was said to have a large heat capacity. The heat capacity of a material is defined as  

The measured value of the heat capacity is found to depend on whether the measurement is made at 

Understanding solids the science of materials. Richard J. D. Tilley 2004 John Wiley Sons, Ltd ISBNs 0 470 85275 5 (Hbk) 0 470 85276 3 (Pbk) 

The heat capacity is an indicator of how much heat has to be added to a unit mass of plastic in order to raise its temperature 1°C, and it is readily measured by ASTM tests. 

Whereas heat capacity is a measure of the energy, thermal diffusivity is a measure of the rate at which energy is transmitted through the plastic it relates to processability. In contrast, metals have values that are hundreds of times larger than those of plastics. 

Source Adapted from Baughman, G.L., Synthetic Fuels Data  

Source Moisture Volatile Matter Fixed Carbon Ash 

The heat capacity is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree. 

The value of the heat capacity depends on the conditions in which the experiment is carried out. The most commonly used heat capacities are  

The isothermal compressibility of water, denoted kj, is a measure of the response of the volume of a liquid to increasing the pressure. This is defined by 

Normally, as the temperature of a liquid increases the average intermolecular distance between the particles of the liquid increases. This makes it easier to compress a liquid at a higher temperature. Thus, we expect the compressibility to increase as we increase the temperature. Water is anomalous in that it has a region below approximately 45°C, where kt actually decreases as the temperature increases (Fig. 1.26). At 45°C, the compressibility passes through a minimum, and thereafter it increases with T, as in the case of normal liquids. 

Liquid water has many other properties that exhibit anomalous values, e.g. dielectric constant, surface tension, diffusion coefficient, viscosity, etc. Aqueous solutions also exhibit unusual properties, some of which will be discussed in Chapter 3. In this introduction, we have mentioned only a few properties, the interpretation of which can be given by some simple theoretical arguments. We shall discuss these properties again in Chapter 2. 

The heat capacity ( specific heat ) Cp of macromolecular substances at constant pressure is the only heat capacity readily accessible experimentally. For theoretical considerations, however, the heat capacity at constant volume Cy is important. According to the laws of thermodynamics, these two quantities are related to each other via the cubic expansion coefficient a and the isothermal compressibility k  

The molar heat capacity of crystalline polymers at constant volume Cy can be theoretically calculated when the frequency spectrum is known. Atoms oscillate harmonically about their equilibrium positions in the crystalline state. In accordance with the Einstein function, each individual oscillation contributes 

At very low temperatures, these lattice oscillations comprise almost all of the heat capacity. At higher temperatures, a correction for the in-harmonicity of the lattice oscillations must be considered. In addition, contributions from group oscillations and rotations about main-chain bonds must also be added at higher temperatures. Finally, a contribution from defects may also be needed. 

The heat capacity of a closed system is defined as the ratio of an infinitesimal quantity of heat transferred across the boundary under specified conditions and the resulting infinitesimal temperature change  

Since g is a path function, the value of the heat capacity depends on the specified conditions, usually either constant volume or constant pressure. Cv is the heat capacity at constant volume and Cp is the heat capacity at constant pressure. These are extensive state functions that will be discussed more fully in Sec. 5.6. 

The heat capacity of a substance is defined as the quantity of heat required to raise the temperature of that substance by one degree on a unit mass (or mole) basis. The term specific heat is frequently used in place of heat capacity. This is not strictly correct, because specific heat has been defined traditionally as the ratio of the heat capacity of a substance to the heat capacity of water. However, since the heat capacity of water is approximately 1 cal/g °C or 1 Btu/lb °F, the term specific heat has come to imply heat capacity. 

For gases, the addition of heat to cause the 1° temperature rise may be accomplished either at constant pressure or at constant volume. Since the amounts of heat necessary are different for the two cases, subscripts are used to identify which heat capacity is being used—Cp for constant pressure and Cy for constant volume. For liquids and solids, this distinction does not have to be made since there is little difference in value between the two. Values of heat capacity are available in the literature.  

Heat capacities are often used on a molar basis instead of a mass basis, in which case the units become cal/gmol - °C or Btu/lbmol - °F. To distinguish between the two bases, uppercase letters (Cp, Cy) are used in this text to represent the molar-based heat capacities, and lowercase letters (cp, cy) are used for the mass-based heat capacities or specific heats. 

Heat capacities are functions of both the temperature and pressure, although the effect of pressure is generally small and is neglected in almost all engineering calculations. The effect of temperature on Cp can be described by 

By definition, the heat capacity of a system is the amount of energy required to raise its temperature by 1 K. The unit is J KT1. To allow calculations and comparisons, the specific heat capacity is more commonly used  

Water has a relatively high specific heat capacity, whereas inorganic compounds have lower heat capacities. Organic compounds are in the medium range (Table 2.2). 

The specific heat capacity of a mixture can be estimated from the specific heat capacities of the different compounds by a mixing rule  

The heat capacity increases with temperature, for example, for liquid water at 20 °C the specific heat capacity is 4.182 kj kg 1 K 1 and at 100 C is 4.216k kg 1K 1 [2]. Its variation is frequently described by the polynomial expression (virial equation)  

whist being toted, no W work is done by a system ai test, nearly all the heat energy goes lino Increasing the temperature. When the system Is allowed to expand at constant pressure, some energy leaves the system as work and the temperature increase is diminished. Thus constant pressure heat capacities are greater than constant volume heal capacities. 

Heat capacity C is a measure of the energy change needed to change the temperature of a substance. The heat capacity is defined as  

Heat capacity is always positive on the MCAT the temperature will always increase when energy is added to a substance at constant volume or pressure. In the real world, heat capacity also changes with temperature tire amount of energy that a substance can absorb per change in temperature varies with the temperature. However, unless otherwise indicated, for the MCAT, assume that heat capacity does not change with temperature. 

Sometimes the MCAT will give you the heat capacity of an entire system. For instance, a thermometer may be made from several substances each with its own heat capacity. The thermometer may be immersed in a bath of oil. The oil has its own heat capacity. On the MCAT, the heat capacity of the thermometer-oil system may be precalculated and given in units of energy divided by units of temperature i.e. J/K or cal/°C. For such a situation, we would use the following equation  

Sometimes the MCAT will give a specific heat capacity c. Specific means divided by mass, so the specific heat capacity is simply the heat capacity per unit mass. A specific heat usually has units of J kg-1 K 1 or cal g 1 °C-. When a specific heat is given, use the following equation  

In this section, we show that the heat capacity (here, at constant volume) is not expressible in terms of the pair distribution function. In fact, we shall see that molecular distribution functions of up to order four are required for this purpose. In Chapter 5, we discuss a different possibility of expressing the heat capacity in terms of generalized molecular distribution functions. 

Consider a system characterized by the variables T, F, N. The internal energy (see Section 3.3) is given by 

The heat capacity is obtained by direct differentiation with respect to temperature 

we have expressed only the nonideal part of the entropy as an average of In P(R ). In fact, this is a general form for the total entropy of a system in any ensemble, i.e., if Pi is the probability of observing the state /, then the entropy is given by S = —k LiPiln Pi. [See, for example. Hill (I960).] 

In the last form on the rhs of (3.108), we have used a common shorthand notation, where the numbers in brackets stand for the configuration of the corresponding particle. We see that the heat capacity is expressible 

The following properties belong to the calorimetric category (1) specific and molar heat capacities, (2) latent heats of crystallization or fusion. It will be shown that both groups of properties can be calculated as additive molar quantities. Furthermore, starting from these properties the molar entropy and enthalpy of polymers can be estimated. 

The specific heat capacity is the heat that must be added per kg of a substance to raise the temperature by one Kelvin or one degree Celsius. The molar heat capacity is the specific heat multiplied by the molar mass (the molar mass of a structural unit in the case of polymers). Specific and molar heat capacity may be defined at constant volume or at constant pressure. The heat added causes a change in the internal energy (It) and in the enthalpy (heat content, H) of the substance. The following notations can be formulated  

Molar heat capacity of solid and liquid polymers at 25°C  

Reliable values for the molar heat capacity in the solid and the liquid state are available for a limited number of polymers only. This emphasizes the importance of correlations between Cp(298) and Cp(298) and the structure of polymers. 

For compounds of low molar mass, correlations are available. Satoh (1948) proposed a method for the prediction of between Cp at 200, 300 and 400 K by the addition of group contributions. Shaw (1969) used the same method for Cp(298) whereas Johnson and Huang (1955) used it for Cp(293). The question was whether these increments are applicable to polymers. 

This rule was stated in 1819 by Dulong and Petit, and it indicates that the specific heat of a metal multiplied by the atomic weight is a constant. This relationship provides a way to estimate the atomic weight of a metal if its specific heat is known. How well the rule holds is indicated by the specific heats of metals shown in Table 7.9. 

For a harmonic oscillator, the mean energy is given in terms of the frequency, v, by 

This can be considered as the average energy of a specific atom over time or the average energy of all the atoms at a specific time. It is useful to consider two special cases. At low temperature, hv kT and the average energy is approximately 1/2hv. At high temperature, hv kT, so e w kT becomes approximately equal to 1 + hv/kT, so that 

This is the classical limit because the energy levels expressed in terms of hv are much smaller than the average energy of the oscillator. 

When the temperature is such that hv kT, neither of the limiting cases described earlier can be used. For many solids, the frequency of lattice vibration is on the order of 1013 Hz, so that the temperature at which the value of the heat capacity deviates substantially from 3R is above 300 to 400 K. For a series of vibrational energy levels that are multiples of some fundamental frequency, the energies are 0, hv, 2hv, 3hi/, etc. For these levels, the populations of the states (n0, nu n2, etc.) will be in the ratio 1 e hl T e Jh,/Ikr e etc. The total number of vibrational states possible for N atoms is 3N 

When a substance is heated, its temperature typically rises. However, the change in temperature, AT, depends on the heat capacity of the substance. The heat capacity, C, is defined as 

If the heat capacity of a beaker of water is 0.50 kj K and we observe a temperature rise of 4.0 K, then we can infer that the heat transferred to the water is 

Heat capacities occur in many places in the following sections and chapters, and we need to be aware of their properties and how their values are reported. First, we note that the heat capacity is an extensive property 2 kg of iron has twice the heat capacity of 1 kg of iron, so twice as much heat is required to change its temperature to the same extent. It is more convenient to report the heat capacity of a substance as an intensive property. We therefore use either the specific heat capacity, Q, the heat capacity divided by the mass of the sample (Cs = C/m, in joules per kelvin per gram, J K g ), or the molar heat capacity, C the heat capacity divided by the amount of substance (C = CIn, in joules per kelvin per mole, J K moT ). In common usage, the specific heat capacity is often called simply the specific heat. 

For reasons that will be explained shortly, the heat capacity of a substance depends on whether the sample is maintained at constant volume (like a gas in a sealed vessel) as it is heated or whether the sample is maintained at constant pressure (like water in an open container) and free to change its volume. The latter is a more common arrangement, and the values given in Table 1.1 are for the heat capacity at constant pressure, Cp. The heat capacity at constant volume is denoted Cy 

The high heat capacity of water is ecologically advantageous because it stabilizes the temperatures of lakes and oceans a large quantity of energy must be 

The thermal properties that are of imporfance in ceramic applications are heat capacity, thermal expansion, and thermal conductivity. Heat capacity is a measure of the heat required for changing fhe femperature. It is therefore important in the heat treatment and use of ceramics. The economy of running a furnace is decided by the amount of heaf needed to reach a particular temperature. Thermal expansion becomes important in ceramic composites. The different constituents in a composite should have close thermal expansion coefficients in order to prevent accumulation of fhermal sfresses in the composite. Thermal conductivity is an undesirable property in the case of insulators. Generally, ceramics possess low thermal conductivity. 

Heat capacity is defined as the heat required to raise the temperature of 1 g of the material by 1°C in centigrade-gram-second (CGS) system. When 1 mole of fhe substance is considered, it is called molar heat capacity. Depending on whether the measurement is taken at constant volume 

E is the internal energy, and H is the enthalpy of the material. The difference between the two heat capacities is given by  

Before the advent of low-temperature physics it was generally accepted that the heat capacity would be independent of the temperature according to the law of Dulong and Petit. Only a few remarkable exceptions were known, such as the heat capacity 

Classical statistical mechanics yields for the energy U of N atoms  

For the thermal properties of solids, Einstein developed an equation that could predict the heat capacity of solids in 1907. This model was then refined by Debye in 1912. Both models predict a temperature dependence of the heat capacity. At 

Pierre Louis Dulong bom Feb. 12, 1785, in Rouen, France, died Jul. 18, 1838, in Paris. 

Alexis Therese Petit, bom Oct. 2, 1791, in Vesoul, died Jun. 21, 1820, in Paris. 

Phase Cp = a + bT + cT, cal/mole deg Accu- racy ( ).% Temper- ature interval. Ref. Year Heat capacity at 20 Cp. cal/mole deg Ref. Year Remarks 

We remarked earlier that heat is often viewed in relation to its effect on the object to which or from which it is transferred. This is the origin of the idea that a body has a capacity for heat. The smaller the temperature change in a body caused by tlie transfer of a given quantity of heat, the greater its capacity. Indeed, a heat capacity might be defined  

The difficulty with tliis is tliat it makes C, like Q, a process-dependent quantity rather than a state function. However, it does suggest the possibility tliat more than one useful heat capacity might be defined. In fact two heat capacities are in common use for homogeneous fluids although tlieir names belie tlie fact, both are state fimctions, defined unambiguously in relation to otlier state fimctions. 

This definition accommodates both the molar heat capacity and the specific heat capacity (usually called specific heat), depending on whether U is the molar or specific internal energy. Although this definition makes no reference to any process, it relates in an especially simple way to a constant-volume process in a closed system, for which Eq. (2.16) may be written  

The combination of this result with Eq. (2.10) for a mechanically reversible, constant-volume process gives  

If tlie volume varies during the process but returns at the end of the process to its initial value, the process cannot rightly be called one of constant volume, even tliough Vi = Vi and A y = 0. However, changes in state functions or properties are independent of path, and are tlie same for all processes which result in the same change of state. Property changes are tlierefore 

Equation 7.17 shows that the total kinetic energy of one mole of a monatomic gas is E = 3 R T/2. Unfortunately the total energy of a system is a difficult quantity to measure directly. It is much easier to measure heat capacities—for example, the number of joules necessary to raise the temperature of one mole of gas by one degree Kelvin. 

Equation 3.12, the work-energy theorem, can be converted into a more convenient form for a gas in a piston in the geometry, as we showed in Equation 3.14  

If the volume is held constant, the integral vanishes. The added energy required to heat one mole of gas from temperature T to temperature T2 (q( T2) — q T )), and the constant-volume molar heat capacity cv, are given by  

The subscripted V is there to remind us that the volume is assumed to remain constant. If the pressure is instead held constant (at the external pressure), the work done from 

Thus the constant-pressure heat capacity cp is given by 

The basis for all calorimetric measurements is the determination of heat capacities. In the absence of any other transition, the DSC curve represents the change in the heat capacity of the sample over the experimental temperature range [5,24]. Detailed descriptions of experimental procedures and data treatment for using DSC to measure heat capacities are available [1,2,5,25-29] A simplistic approach is given below. 

The heat absorbed on heating a sample with constant heat capacity, Cp, between temperatures T0 and T is, by definition, 

The amount of heat energy associated with a given temperature change in a given system is a function of the chemical and physical states of the system. A measure of this heat energy can be quantified in terms of the quantity known as the heat capacity which may be expressed on a mass or molar basis. The former is designated the specific heat capacity (Jkg K ) and the latter the molar heat capacity (Jmol K ). The relationships between some commonly used heat capacity units are  

For gases two heat capacities have to be considered, at constant pressure, Cp, and at constant volume, Cv. The value of the ratio of these two quantities, Cp/Cv = 7, varies from about 1.67 for monatomic gases (e.g. He) to about 1.3 

Specific heat capacities of solid substances near normal atmospheric temperature can be estimated with a reasonable degree of accuracy by combining two empirical rules. 

The first of these, due to Dulong and Petit, expresses a term called the atomic heat which is defined as the product of the relative atomic mass and the specific heat capacity. For all solid elemental substances, the atomic heat is assumed to be roughly constant  

The second rule, due to Kopp, applies to solid compounds and may be expressed by 

Equations (3.62) and (3.63) require the temperature 5 in °C and the volatile matter VM as weight fraction (wt%(waf)/100) while the heat capacity is calculated in kJ/(kg-K). 

For the ash heat capacity Cash in kJ/(kg-K) munerous correlations exist [2]  

All deliver values in the range of 0.8 to 1.2kJ/(k g-K), which may be distinguished according to the main ash fractions either being dominated by calcia and iron oxide or by alumina and silica [2]. 

Finally, the heat capacity of the water must be incorporated to unify all constituents in one value for the coal on as-received basis. 

In Equation (3.66), W and A denote the weight fractions of water and ash (wt% (ar)/100) and ChjO is the heat capac% of water, which is 4.187 kJ/(kg-K). Attention must be paid to the vaporization temperature, which depends on the pressure. If it is exceeded during a certain process, the enthalpy of vaporization must be incorporated in the calculations and the water fraction is correspondingly reduced. 

Imagine that you have a pure liquid and you wish to calculate how much energy is required to heat that liquid from one temperature to another. This calculation is simple, providing you know a property of the material known as the heat capacity (or specific heat). There are two types of heat capacities that can be defined. 

The heat capacity at constant temperature (Cp) and the heat capacity at constant volume (C ), which are defined as follows 

Reference has already been made to the range of applications of DTA and DSC. This section is concerned with a closer look at calorimetric measurements which link thermal power to heat capacity, dAq/dt = (Cg — and its integral, dAq/dt)dt, to energy or enthalpy. These linkages together with temperature form the basis of quantitative DSC. Computer systems for the control of equipment, data capture and subsequent analysis have combined to give increased versatility and results of far greater precision than was possible previously with chart recorders. 

This represents one of the early successes of conventional DSC and more recently of MTDSC. Heat capacity is a key thermodynamic quantity because of its intrinsic importance and its relationship to other quantities - enthalpy, entropy and Gibbs energy. Its measurement continues to be an important application of DSC where results can be obtained having an uncertainty of only 1-2%, often with a minimum of difficulty. At the other end of the scale with rigorous attention to detail and with suitable samples results can be obtained with uncertainties approaching those of adiabatic calorimetry.  

The difference between the signals for experiments 1 and 2 serves to calibrate the thermal power, 

The heat capacity of the sample is obtained from the formula, 

It is seen that the calibration constant disappears, which assumes that it is constant over the experimental conditions. The calculation is carried out using dedicated software. In some circumstances the crucible used for the sample may have to be different from that used for the calibrant. This means that a correction will be required to take into account the difference between the heat capacity of the two crucibles - readily calculated with sufficient accuracy. Measurements can be made at a series of temperatures but are meaningful only within the quasi-steady-state region of the experiment. The specific heat capacity of sapphire has been listed by ASTM in connection with the standard test method E 1269 (1999) for determining specific heat capacity by differential scanning calorimetry. 

However, to estimate the standard enthalpy of formation, it is necessary to add two reactions to Equation (4.46), because, by definition, the standard enthalpy of formation refers to the formation of the compound in its standard state from the elements in their standard states. Therefore we introduce the following enthalpy changes to convert the elements from their standard states to the gaseous atoms at 298 K  

If we wish to know the enthalpy of formation of liquid Se2Cl2, we can estimate the enthalpy of condensation (perhaps from Trouton s rule [13] or by comparison with related sulfur compounds) and can add it to the value of A// , obtained in Equation (4.49). 

By these methods, one can obtain fairly reliable estimates of enthalpies of formation of many compounds. As the bond enthalpies used are average values, they cannot be expected to result in highly accurate results for enthalpies of formation. More complex procedures also have been developed that will provide greater accuracy [14]. Related methods for estimation of thermodynamic data are discussed in Appendix A. 

We introduced the enthalpy function particularly because of its usefulness as a measure of the heat that accompanies chemical reactions at constant pressure. We will find it convenient also to have a function to describe the temperature dependence of the enthalpy at constant pressure and the temperature dependence of the energy at constant volume. Eor this purpose, we will consider a new quantity, the heat capacity. (Historically, heat capacity was defined and measured much earlier than were enthalpy and energy.) 

Fundamental Statement. The heat absorbed by a body (not at a transition temperature) is proportional to the change in temperature  

Your objectives in studying this section are to be able to  

-Gonvert-an-expression-for-the-heat-capacity-from-one-set-of units-to- 

Look up from a reference source an equation that expresses the heat capacity as a function of temperature, and compute the heat capacity at a given temperature. 

Fit empirical heat capacity data with a suitable function of temperature by estimating the values of the coefficients in the function. 

The system expands as the temperature increases at constant pressure. Work is needed against the molecules and the atmosphere as they move apart. In the critical region the coefficient of thermal expansion at constant pressure is large and thus the effect on Cp is also large. 

Because of the great difference in the thermal expansion of polymers and metals or glass, significant problems can arise when thermal stress is applied to composites of these materials. The so-called dimensional stability of the polymer is also of technological importance. Dimensionally stable polymers must not only possess a small coefficient of thermal expansion They should also not exhibit recrystallization phenomena. Recrystallizations lead to distortions because of the difference in densities between amorphous and crystalline regions. 

Of course this is the expression for the vibrational energy relative to the first level, and tends to zero as temperature tends to zero. But the absolute value of the vibrational energy is never zero (recall equation 3.24). For an order of magnitude evaluation, the zero-point energy l/2hv for a 100 cm frequency, a typical value for a lattice vibration, is about 0.5 kJ mol .  

4 Internal energy II From thermal and mechanical experiments 

These two equations were in fact proposed long before the existence and the properties of molecules were discovered, solely on the basis of thermal and mechanical experiments on macroscopic systems. Conceptually, the key of the first principle of thermodynamics is not in the balance of equation 7.26, but in the fact that (as recognized by Joule) (/ is a state function, that is, its value in a given state is always the same, no matter how that state has been reached or what kind of energy is exchanged. This is quite understandable in the light of equation 7.22, which only depends on energy levels of the system as it is, not as it has ever been. 

There is no way of directly measuring absolute values of internal energies, and there is no way of calculating them either, because V is not known. But suppose a controlled 

In equation 7.30, index k runs on all normal modes of vibration, and (3natom - 5)7 holds for linear molecules. 

Note We have not made any assumption on the type of energy carriers. Hence, this is a universal law for all energy carriers. Only assumption is made about local thermodynamic equilibrium. Neglecting photon contribution, the thermal conductivity can be written as 

Heat capacity of a material is defined as the change in internal energy resulting from a change in temperature. The energy within a crystalline material is stored in the free electrons of a metal and within the lattice in the form of vibrational energy. 

Note the primary mechanism by which thermal energy is assimilated in solid materials. 

Determine the linear coefficient of thermal expansion, given the length alteration that accompanies a specified temperature change. 

Briefly explain the phenomenon of thermal expansion from an atomic perspective using a 

Note the two principal mechanisms of heat conduction in solids, and compare the relative magnitudes of these contributions for each of metals, ceramics, and polymeric materials. 

Thermal property refers to the response of a material to the apphcation of heat. As a solid absorbs energy in the form of heat, its temperatnre rises and its dimensions increase. The energy may be transported to cooler regions of the specimen if temperature gradients exist, and nltimately, the specimen may melt. Heat capacity, thermal expansion, and thermal condnctivity are properties that are often critical in the practical use of solids. 

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Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

The enthalpy of a vapor mixture is obtained first, from zero-pressure heat capacities of the pure components and second, from corrections for the effects of mixing and pressure. 

The heat capacity of an ideal vapor is a monotonic function of temperature in this work it is expressed by the empirical relation... 

This chapter presents quantitative methods for calculation of enthalpies of vapor-phase and liquid-phase mixtures. These methods rely primarily on pure-component data, in particular ideal-vapor heat capacities and vapor-pressure data, both as functions of temperature. Vapor-phase corrections for nonideality are usually relatively small. Liquid-phase excess enthalpies are also usually not important. As indicated in Chapter 4, for mixtures containing noncondensable components, we restrict attention to liquid solutions which are dilute with respect to all noncondensable components. 

Appendix C-3 gives constants for the ideal-gas, heat-capacity equation... 

Reactor heat carrier. Also as pointed out in Sec. 2.6, if adiabatic operation is not possible and it is not possible to control temperature by direct heat transfer, then an inert material can be introduced to the reactor to increase its heat capacity flow rate (i.e., product of mass flow rate and specific heat capacity) and to reduce... 

Consider the simple flowsheet shown in Fig. 6.2. Flow rates, temperatures, and heat duties for each stream are shown. Two of the streams in Fig. 6.2 are sources of heat (hot streams) and two are sinks for heat (cold streams). Assuming that heat capacities are constant, the hot and cold streams can be extracted as given in Table 6.2. Note that the heat capacities CP are total heat capacities and... 

Stream Type Supply temp. Ts CO Target temp. Tr (°C) AH (MW) Heat capacity flow rate CP (MW°C )... 

Taking the heat capacity of water to be 4.3 kJ kg K , heat duty on boiler feedwater preheating is... 

The Ft correction factor is usually correlated in terms of two dimensionless ratios, the ratio of the two heat capacity flow rates R and the thermal effectiveness P of the exchanger ... 

Example 9.1 A process involves the use of benzene as a liquid under pressure. The temperature can be varied over a range. Compare the fire and explosion hazards of operating with a liquid process inventory of 1000 kmol at 100 and 150°C based on the theoretical combustion energy resulting from catastrophic failure of the equipment. The normal boiling point of benzene is 80°C, the latent heat of vaporization is 31,000 kJ kmol the specific heat capacity is 150 kJkmoh °C , and the heat of combustion is 3.2 x 10 kJkmok. ... 

Heat carriers. If adiabatic operation produces an unacceptable rise or fall in temperature, then the option discussed in Chap. 2 is to introduce a heat carrier. The operation is still adiabatic, but an inert material is introduced with the reactor feed as a heat carrier. The heat integration characteristics are as before. The reactor feed is a cold stream and the reactor efiluent a hot stream. The heat carrier serves to increase the heat capacity fiow rate of both streams. 

Stream Supply - temperature (°C) Target temperature (°C) J. Heat capacity flow rate (MW°C b... 

Exampie A.3.1 The pressures for three steam mains have been set to the conditions given in Table A.l. Medium- and low-pressure steam are generated by expanding high-pressure steam through a steam turbine with an isentropic efficiency of 80 percent. The cost of fuel is 4.00 GJ and the cost of electricity is 0.07 h. Boiler feedwater is available at 100°C with a heat capacity... 

R distillation column reflux ratio (-) or heat capacity ratio of 1-2 shell-and-tube heat exchanger (-)... 

This definition is in terms of a pool of liquid of depth h, where z is distance normal to the surface and ti and k are the liquid viscosity and thermal diffusivity, respectively [58]. (Thermal diffusivity is defined as the coefficient of thermal conductivity divided by density and by heat capacity per unit mass.) The critical Ma value for a system to show Marangoni instability is around 50-100. 

Brunauer and co-workers [129, 130] found values of of 1310, 1180, and 386 ergs/cm for CaO, Ca(OH)2 and tobermorite (a calcium silicate hydrate). Jura and Garland [131] reported a value of 1040 ergs/cm for magnesium oxide. Patterson and coworkers [132] used fractionated sodium chloride particles prepared by a volatilization method to find that the surface contribution to the low-temperature heat capacity varied approximately in proportion to the area determined by gas adsorption. Questions of equilibrium arise in these and adsorption studies on finely divided surfaces as discussed in Section X-3. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

The integral can be approximated by noting that the derivative of the Femii function is highly localized around E. To a very good approximation, the heat capacity is... 

As one raises the temperature of the system along a particular path, one may define a heat capacity C = D p th/dT. (The tenn heat capacity is almost as unfortunate a name as the obsolescent heat content for// alas, no alternative exists.) However several such paths define state functions, e.g. equation (A2.1.28) and equation (A2.1.29). Thus we can define the heat capacity at constant volume Cy and the heat capacity at constant pressure as... 

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