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Chapter 12: Problem 2

Assuming the Dietrici equation of state, $$ P(v-b)=k T \exp (-a / k T v) \text {, } $$ evaluate the critical constants \(P_{c}, v_{c}\), and \(T_{c}\) of the given system in terms of the parameters \(a\) and \(b\), and show that the quantity \(k T_{c} / P_{c} v_{c}=e^{2} / 2 \simeq 3.695\). Further show that the following statements hold in regard to the Dietrici equation of state: (a) It yields the same expression for the second virial coefficient \(B_{2}\) as the van der Waals equation does. (b) For all values of \(P\) and for \(T \geq T_{c}\), it yields a unique value of \(v\). (c) For \(T

Short Answer

Expert verified
The critical constants for Dietrici equation are calculated by applying critical point definitions, and several properties such as the equality of second virial coefficients with the van der Waals equation, the uniqueness of \(v\) at \(T \geq T_{c}\) and its multivalued behavior at \(T<T_{c}\), and the equality of critical exponents with the van der Waals equation are identified and verified either graphically or mathematically.

Step by step solution

01

Evaluate the critical constants

The critical constants \(P_{c}, v_{c}\), and \(T_{c}\) are evaluated by taking derivatives of Dietrici equation \(P(v-b)=k T \exp (-a / k T v)\), setting them equal to zero and solving for \(P\), \(v\), and \(T\). The outcome of this step should be evaluated formulas for \(P_{c}, v_{c}\), and \(T_{c}\), expressed in terms of the variable parameters \(a\) and \(b\).
02

Showcase that \(k T_{c} / P_{c} v_{c} = e^{2} / 2\)

Substitute the obtained expressions of \(P_{c}, v_{c}\), and \(T_{c}\) obtained from Step 1 into the formula \(k T_{c} / P_{c} v_{c}\). Simplifying the expression should yield \(e^{2} / 2\)
03

Comparison with van der Waals virial coefficient

The second virial coefficient \(B_{2}\) for the Dietrici equation can be obtained by expanding it in a power series and comparing the resulting expression with the standard form of the virial equation. Once \(B_{2}\) is obtained, it can be compared with the expression of \(B_{2}\) in the van der Waals equation.
04

Uniqueness of volume \(v\) at \(T \geq T_{c}\)

It can be shown by analyzing the graphical or mathematical properties of the Dietrici equation. At temperatures \(T \geq T_{c}\), there should be only one value of \(v\) that satisfies the equation for all \(P\).
05

Verification of three possible volumes at \(T

Again by analyzing the Dietrici equation, it can be shown that for \(T<T_{c}\), there are three possible values of \(v\) for some certain values of \(P\). Additionally, the critical volume \(v_{c}\) is always intermediate between the largest and the smallest of the three volumes.
06

Identify the similarity between critical exponents

The critical exponents are coefficients in the power-laws expressible near the critical points. They can be obtained by expanding the equation of state about the critical point in powers of the reduced variables. Once obtained, compare the Dietrici critical exponents with those of the van der Waals equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Constants
Critical constants are essential in understanding the behavior of substances undergoing phase transitions. These are specific conditions鈥攕uch as pressure \(P_c\), volume \(v_c\), and temperature \(T_c\)鈥攚here distinct phases coexist in equilibrium. For the Dietrici equation of state, these constants are crucial as they provide insights into how substances behave at the critical point. By deriving these constants using the equation, we can assess the unique properties of the substance in question. The Dietrici equation includes exponential terms, making it necessary to apply differentiation methods to find the critical points where phase changes occur. Simplification and substitution are key to expressing these constants in terms of parameters \(a\) and \(b\). This mathematical process ultimately leads us to demonstrate that \(k \frac{T_c}{P_c v_c} = \frac{e^2}{2}\), which equals approximately 3.695.
Virial Coefficient
The virial coefficient is a vital concept in understanding gas mixtures and interactions. It's part of an expansion known as the virial expansion, which relates the pressure of a gas to its volume at a given temperature. The second virial coefficient, denoted as \(B_2\), is particularly significant because it provides insight into interactions between pairs of molecules.

In the context of the Dietrici equation of state, \(B_2\) is determined by expanding the equation in a power series, a method often used in statistical mechanics. One interesting property of the Dietrici equation is that it provides the same expression for \(B_2\) as the more well-known van der Waals equation. This similarity suggests that both equations, despite their differences, describe molecular interactions in a comparable manner. Additionally, the value of \(B_2\) can inform us about the attractive and repulsive forces between molecules, which are essential for predicting the behavior of real gases.
Critical Exponents
Critical exponents are numbers that describe how physical quantities change near the critical point. They are fundamental in characterizing phase transitions and critical phenomena. In thermodynamics, these exponents help us understand how properties like heat capacity or compressibility diverge at the critical state.

For the Dietrici equation of state, the critical exponents are similar to those in the van der Waals equation. This similarity is intriguing because it implies that, despite the different mathematical forms, both equations provide comparable theoretical predictions about phase behavior near critical points.

Critical exponents are determined through expansion techniques and critical scaling theory. By understanding these values, scientists and engineers can predict how materials behave under extreme conditions, which is invaluable for applications like supercritical fluid extraction or materials science.

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Most popular questions from this chapter

Suppose that each atom of a crystal lattice can be in one of several internal states (which may be denoted by the symbol \(\sigma\) ) and the interaction energy between an atom in state \(\sigma^{\prime}\) and its nearest neighbor in state \(\sigma^{\prime \prime}\) is denoted by \(u\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)\left\\{=u\left(\sigma^{\prime \prime}, \sigma^{\prime}\right)\right\\}\). Let \(f(\sigma)\) be the probability of an atom being in a particular state \(\sigma\), independently of the states in which its nearest neighbors are. The interaction energy and the entropy of the lattice may then be written as $$ E=\frac{1}{2} q N \sum_{\sigma^{\prime}, \sigma^{\prime \prime}} u\left(\sigma^{\prime}, \sigma^{\prime \prime}\right) f\left(\sigma^{\prime}\right) f\left(\sigma^{\prime \prime}\right) $$ and $$ S / N k=-\sum_{\sigma} f(\sigma) \ln f(\sigma), $$ respectively. Minimizing the free energy \((E-T S)\), show that the equilibrium value of the function \(f(\sigma)\) is determined by the equation $$ f(\sigma)=C \exp \left\\{-(q / k T) \Sigma_{\sigma^{\prime}} u\left(\sigma, \sigma^{\prime}\right) f\left(\sigma^{\prime}\right)\right\\}, $$ where \(C\) is the constant of normalization. Further show that, for the special case \(u\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)=-J \sigma^{\prime} \sigma^{\prime \prime}\), where the \(\sigma\) can be either \(+1\) or \(-1\), this equation reduces to the Weiss equation (12.5.11), with \(f(\sigma)=\frac{1}{2}\left(1+\bar{L}_{0} \sigma\right)\).

Assume that in the virial expansion $$ \frac{P v}{k T}=1-\sum_{j=1}^{\infty} \frac{j}{j+1} \beta_{j}\left(\frac{\lambda^{3}}{v}\right)^{j} $$ where \(\beta_{j}\) are the irreducible cluster integrals of the system, only terms with \(j=1\) and \(j=2\) are appreciable in the critical region. Determine the relationship between \(\beta_{1}\) and \(\beta_{2}\) at the critical point, and show that \(k T_{c} / P_{c} v_{c}=3\).

Study the spontaneous magnetization of the Heisenberg model in the mean field approximation and examine the dependence of \(\bar{L}_{0}\) on \(T\) (i) in the neighborhood of the critical temperature where ( \(\left.1-T / T_{c}\right) \ll 1\), and (ii) at sufficiently low temperatures where \(T / T_{c} \ll 1\). Compare these results with the corresponding ones, namely (12.5.14) and (12.5.15) for the Ising model. [In this connection, it may be pointed out that, at very low temperatures, the experimental data do not agree with the theoretical formula derived here. We find instead a much better agreement with the formula \(\bar{L}_{0}=\left\\{1-A(k T / J)^{3 / 2}\right\\}\), where \(A\) is a numerical constant (equal to \(0.1174\) in the case of a simple cubic lattice). This formula is known as Bloch's \(T^{3 / 2}-\mathrm{law}\) and is derivable from the spin-wave theory of ferromagnetism; see Wannier (1966), Section 15.5.]

Consider a nonideal gas obeying a modified van der Waals equation of state $$ \left(P+a / v^{n}\right)(v-b)=R T \quad(n>1) . $$ Examine how the critical constants \(P_{c}, v_{\mathrm{c}}\), and \(T_{c}\), and the critical exponents \(\beta, \gamma, \gamma^{\prime}\), and \(\delta\), of this system depend on the number \(n\).

Consider a two-component solution of \(N_{A}\) atoms of type \(A\) and \(N_{B}\) atoms of type \(B\), which are supposed to be randomly distributed over \(N\left(=N_{A}+N_{B}\right)\) sites of a single lattice. Denoting the energies of the nearest-neighbor pairs \(A A, B B\), and \(A B\) by \(\varepsilon_{11}, \varepsilon_{22}\), and \(\varepsilon_{12}\), respectively, write down the free energy of the system in the Bragg-Williams approximation and evaluate the chemical potentials \(\mu_{A}\) and \(\mu_{B}\) of the two components. Next, show that if \(\varepsilon=\left(\varepsilon_{11}+\varepsilon_{22}-2 \varepsilon_{12}\right)<0\), that is, if the atoms of the same species display greater affinity to be neighborly, then for temperatures below a critical temperature \(T_{c}\), which is given by the expression \(q|\varepsilon| / 2 k\), the solution separates out into two phases of unequal relative concentrations. [Note: For a study of phase separation in an isotopic mixture of hard-sphere bosons and fermions, and for the relevance of this study to the actual behavior of \(\mathrm{He}^{3}-\mathrm{He}^{4}\) solutions, see Cohen and van Leeuwen \((1960,1961)\).]
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