In the expansion process undergone by a parcel (fixed mass) in moving from (VA,pA) to (VB,pB), the parcel will do some work on the environment, /VvB p(V) dV. During this process some energy might be transported into the system because of a temperature difference between the interior of the system and the surrounding environment. This energy transported into the system is thermal energy 2 as described in Chapter 2. Thermal energy consists of all the modes of energy associated with individual molecules: translational kinetic energy, rotational energy for polyatomic molecules and vibrational energy (potential and kinetic) for polyatomic molecules that experience internal stretching oscillations. These individual energy terms each contribute to the thermal energy of a molecule (but only the translational kinetic energy contributes to pressure). In fact, the energy on the average is shared equally between the different modes (translational kinetic, etc.), although at atmospheric temperatures the vibrational modes are not excited because of quantum threshold effects.3
This transport of heat is effected at the molecular level by the collisions of individual molecules. If there is a gradient of temperature, molecules from the warmer region will penetrate a distance of the order of a mean free path before suffering a collision into the cooler region (and vice versa), causing the cooler region to warm; molecules moving the other way cause the warmer region to cool through individual collisions. The random motions of molecules crossing the boundary bring the news of their different "temperature" via a random walk process (each step forward or backward determined by the proverbial flip of a coin). The news and conversion are brought about slowly but surely. The distance advanced by the spreading edge of a "warm front" at the molecular level is proportional to the square root of the time elapsed. This is in stark contrast to the propagation of pressure differences which move via a sound-like wave (distance of advance of the pressure front being proportional to time). To obtain an idea of this contrast consider a one-dimensional gas (x direction only) and let an instantaneous hot spot develop at x = 0 (perhaps a fire cracker explodes). It is possible to solve this problem analytically, but the details need not be given here. The basic idea is that heat flux
2 In most texts this thermal motion is referred to as heat as though it were a material substance moving around in space, but some authors (e.g., Bohren and Albrecht 1998) shun the use of the noun heat in favor of the verbs heating or cooling as a transport process involving the energizing of neighboring molecules by their aggregate being in contact with an aggregate of molecules of a different temperature. We will use the term heat to mean the integral over the heating rate with respect to time. Just keep in mind that heat is not a fluid flowing about in the medium.
3 The energy levels in quantum mechanics are discrete and the disturbing collisions need to have a sufficient energy transfer to effect a transition to the next higher energy level. Typically, rotational levels are closely enough spaced for them to be excited, but vibrational thresholds are much higher, requiring very high temperatures for excitation.
Figure 3.4 Spread in meters of a localized pulse of thermal energy due (only) to molecular thermal conductivity in air at STP after 2 h and after 12 h. The functional form is the normal distribution with standard deviation a = V2Dt, where D = KH/pcp « 2 x 10-5 m 2 s-1 and t is time in seconds.
is proportional to — KHdT/dx, where kh is the thermal conductivity of air (~0.024 Jm-1s-1K-1). The pulse spreads out in the shape of a normal distribution as shown in Figure 3.4. The standard deviation of the spread of the elevated temperatures is only about 3 m after 12 hours. This means that the concept of parcel integrity for objects of the order of several hundred meters is safe for days if the only stirring mechanism is molecular diffusion. There are other mechanisms that can shorten the time of mixing depending on the conditions, but these are still usually slow compared to the adjustment of the interior to the exterior pressure. Note that sound waves travel at several hundred meters per second. The adjustment of pressures should be accomplished in several hundred passes of sound waves back and forth across the parcel - still very fast compared to molecular and even eddy (turbulent) transport processes. The sound waves are eventually dissipated into thermal energy.
Thermal energy or heat as we have been discussing it can now be contrasted with the work being done by a system during a process. The thermal energy is at the molecular level and it migrates from place to place via gradients in the temperature (say from the system to a reservoir), while work is at the truly macroscopic level. Work is performed when one of the macroscopic dimensions (sometimes called configuration coordinates), say the position of a piston, is altered a finite (macroscopic) amount.
Returning to our system, heat can be transported into it because of small differences in temperature between the system and its environment. The amount of heat taken into the system during a finite process is traditionally given the symbol Q. We say 4 QA^B as the system moves from the state denoted A to the state denoted B.
4 Note that in our notation if a positive amount of heat is absorbed by the system then Qa^B is assigned a positive value. This is the sign convention followed by virtually all textbooks.
Consider first the simple heating of a parcel where the volume is held fixed (an isochoric process). The parcel is heated by an amount QA^B and its temperature undergoes a change AT = TB - TA. The heat energy absorbed and the temperature change are related with the coefficient of proportionality being the mass times the specific heat capacity cv (units J kg-1 K-1); Cv = Mcv, where Cv (units J K-1) is called the total heat capacity:
In general, cv might be a function of temperature, but for an ideal gas it is not. For dry air the value of cv is 717 J kg-1 K-1. In this process no work is done by the parcel on its environment, since the volume of the parcel does not change. All the heat_given to the system goes into its internal energy. From the molecular relation 1 mv2 = |kBT we recall that the average kinetic energy of the molecules is proportional to the Kelvin temperature. Hence, a change in internal energy is equivalent to a change in the kinetic energy (for an ideal monatomic gas). As we will see later the kinetic energy of translation still has the same relation to temperature for multi-atomic gases, but the internal energy in the multi-atomic case is modified (next section).
In chemical applications (also chemical texts and handbooks) it is common to use the molar specific heat for a substance, cv. In this case the total heat capacity is Cv = vcv where v is the number of moles in the system. In this formulation:
Example 3.3 A sealed room with walls made of perfectly insulating material has dimensions 4mx4mx3m. The conditions are p = 1000 hPa, T = 300 K. What is the mass of air in the room? How many joules are required to raise the temperature by 1 K?
Answer: M = pV/RdT = (105Pa x 48m3)/(287 Jkg-1 K-1 x 300K) = 55.7 kg.
Q = cvMAT = 717 Jkg-1 K-1 x 55.7kg x 1K = 4.0x105J = 400kJ. □
Example 3.4 How many kilowatt hours of energy are expended in the last example? Answer: 1 kWh = 3600 kJ. So the result is 0.11 kWh. Note that the cost of 1 kWh is a few cents (US). □
Example 3.5 In the last example, consider the effect of thin walls. Let us take the walls to be wood and 1 cm thick. What is the amount of heat necessary to bring these walls (and floor) up 1°C?
Answer: The specific heat of wood is 1760 J kg-1 K-1, and the density of wood is 600 kg m-3. The walls, ceiling and floor have an area of 72 m2 making a volume of 0.72 m3. The mass of this material is 600 kg m-3 x 0.72 m3 = 432 kg. The total heat capacity of the solid matter is 760 kJ, nearly twice that of the air contained. □
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