Heat capacity Einstein equation

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

Figure 10.12 Comparison for diamond of the experimental Cr.m (circles) and the prediction of the Einstein heat capacity equation with = 1400 K (solid line). The experimental results below T = 300 K are closely spaced in temperature, and not all are shown in the figure.

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

Einstein heat capacity equation 569-72 Schottky effect 580—5 solid + solid phase transitions 399-404 first-order 402-4 solutes 6... 

Debye heat capacity equation 572-80 Einstein heat capacity equation 569-72 heat capacity from low-lying electronic levels 580-5 Schottky effect 580-5 statistical weight factors in energy levels of ideal gas molecule 513 Stirling s approximation 514, 615-16 Streett, W. B. 412... 

This is, of course, the reciprocal of the time taken for an atom to move a lattice site distance into a vacancy. um can be estimated from the heat capacity of the crystalline material using the Einstein or Debye models6 of atoms as harmonic oscillators in a lattice. Combining Equations (2.30) and (2.31) gives the number of atoms moving per second as... 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

HEAT CAPACITY EQUATION (Einstein). A quantum relationship for the heat capacity at constant volume of an element of the form ... 

Our discussion of the specific heat capacity of polymers on the preceding pages has been quite empirical. There are, in fact, few fundamental rules that can be used for the prediction of specific heat capacity. At very low temperatures, the equations of Debye and Einstein may be used. 

The melting point of n-hexanol is —47.2°C (225.8 K), and its enthalpy of fusion is 3676 cal/ (g-mol). The heat capacity of crystalline n-hexanol (Cp)erysiai at temperatures below 18.3 K may be estimated using the Debye-Einstein equation ... 

The heat capacity equation (V.81) was used to derive an expression for the entropy change in the temperature range 53 to 1173 K. Below 50 K, Kelley [41KEL] extrapolated a value for the entropy change of 2.34 J-K -mor from the funetion sum (using Debye and Einstein terms) derived between 51 and 298 K the Einstein terms, however, only contribute a negligible amount to the value. The values obtained below and above 50 K were then combined to derive a value for the standard entropy at... 

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... 

In terms of 0e the Einstein equation [8.7] fa- the heat capacity of solids is... 

From Eq. (3) the frequency distribution can be calculated following the Debye treatment by making use of the fact that an actual atomic system must have a limited number of frequencies, limited by the number of degrees of freedom N. The distribution p(v) is thus simply given by Eq. (4). This frequency distribution is drawn in the sketch on the right-hand side in Fig. 2.36. The heat capacity is calculated by using a properly scaled Einstein term for each frequency. The heat capacity function for one mole of vibrators depends only on Vj, the maximum frequency of the distribution, which can be converted again into a theta-temperature, j. Equation (5) shows that at temperature T is equal to R multiplied by the one-dimensional Debye... 

In 1907, Einstein applied energy quantization to the vibrations of atoms in a solid element, assuming that each atom s vibrational energy in each direction x, y, z) is restricted to be an integer times hv n, where the vibrational frequency I vib is characteristic of the element. Using statistical mechanics, Einstein derived an expression for the constant-volume heat capacity Cy of the solid. Einstein s equation agreed fairly well with known Cv-versus-temperature data for diamond. 

Just like gases, solids can lose their ability to absorb energy at low temperatures. According to equipartition Equation (11.54), each vibration contributes kT to the energy. If there are N atoms in a solid, and each atom has three vibrational modes (in the x-, y-, and z-directions), the heat capacity wall be C = 3Nk, independently of temperature. This is called the Law of Dulong and Petit, named after the experimentalists who first observ ed this behavior around 1819. But more recent experimental data, such as that shown in Figures 11.14 and 11.15, indicate that this law does not hold at low temperatures. As the temperature approaches zero, Cv 0. The Einstein model, developed in 1907, shows why. This work was among the first evidence for the quantum theory of matter. 

Notice how the Einstein temperature and the absolute temperature of the crystal always appear together as the fraction Be/T. Notice, too, that there is nothing in equation 18.64 that is sample-dependent other than the Einstein temperature Be-This means that if the heat capacity of any crystal were plotted versus Be/T, all of the graphs would look exactly the same. This is one example of what is called a law of corresponding states. Einstein s derivation of a low-temperature heat capacity of crystals was the first to predict such a relationship for all crystals. 

How do we determine the Einstein temperature Be without knowing the characteristic vibrational frequency of the atoms in the crystal Typically, experimental data is fitted to the mathematical expression in equation 18.64 and a value of the Einstein temperature is used to allow for the best possible fit to experimental results. For example, a plot of experimental measurements ofthe heat capacity versus Tdividedby E (which isproportional to T, whereas 0g/r is inversely proportional to T and less easy to graph as T —> 0 K) is shown in Figure 18.4. Notice that there is reasonable agreement between experiment and theory, suggesting that Einstein s statistical thermodynamic basis of the heat capacity of crystals has merit. Table 18.6 lists a few experimentally determined Einstein temperatures for crystals. 

However, the Einstein equation deviates from experimental values at very low temperatures, predicting a lower heat capacity than is measured experimentally. In fact, using the mathematics of limits, it can be shown that equation 18.64 predicts the following ... 

The integral in equation 18.68 cannot be solved analytically, but its value can be determined numerically. Just like the Einstein treatment of heat capacities of crystals, the Debye temperature do is selected so that the numerical evaluation of equation 18.68 agrees as closely as possible with experimental data. Figure 18.6 shows what Debye s equation looks like and Table 18.7 lists some values of do-Applying limits to equation 18.68 shows that... 

Silver metal is a very good conductor of heat. The following are heat capacities at different temperatures. Using equation 18.64, determine a value for the Einstein temperature that best fits this data. 

C) Einstein s equation for the molar heat capacity of a solid, Cy (in joules/(mol K)),... 

Use the heat capacity data of Problem 3 to estimate the Einstein e temperature for Cu (see Fig. 5.13 for equations, or use a table of the Einstein function). 

This relation, commonly known as Einstein s (1905) law of heat capacities, correctly corresponds to the limiting cases Cy = Oat T = 0 and to Cy = 3R at infinite temperature—the law of Dulong and Petit. It immediately rationalizes the deviations of the observed Cy values from this law at lower temperatures that had mystified scientists before the advent of quantum mechanics. However, the above equation generally deviates to some extent from experiment in particular, at low T, the above relation shows that Cy, while experiments conform to aCy T variation. 

In Fig. 9.5, the Einstein model heat capacity is shown as a function of the characteristic temperature. Equation 9.69 correctly predicts the heat capacity at the limits of T 0 and T oo. 

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