/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Assume that in the virial expans... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 1

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 \(k T_{c} / P_{c} v_{c}=3\).

Short Answer

Expert verified
To achieve \(k T_{c} / P_{c} v_{c} = 3\), we choose \(\beta_{1}\left(\frac{\lambda^{3}}{v_{c}}\right)\) and \(\beta_{2}\left(\frac{\lambda^{3}}{v_{c}}\right)^2\) in such a way that they are equal. The established relationship between the cluster integrals at the critical point allows for simplification and satisfying the given condition.

Step by step solution

01

Simplify Virial Expansion

Initially, the given virial expansion is\[\frac{P v}{k T}=1-\sum_{j=1}^{\infty} \frac{j}{j+1} \beta_{j}\left(\frac{\lambda^{3}}{v}\right)^{j}\]Considering only terms with \(j = 1\) and \(j = 2\), the equation reduces to\[\frac{P v}{k T}=1-\frac{1}{2}\beta_{1}\left(\frac{\lambda^{3}}{v}\right)-\frac{2}{3} \beta_{2}\left(\frac{\lambda^{3}}{v}\right)^2\]
02

Evaluate at Critical Point

At the critical point, \((P_{c}, v_{c}, T_{c})\), substituting \(P v / k T\) with \(P_{c} v_{c} / k T_{c}\) gives\[\frac{P_{c} v_{c}}{k T_{c}}=1-\frac{1}{2}\beta_{1}\left(\frac{\lambda^{3}}{v_{c}}\right)-\frac{2}{3} \beta_{2}\left(\frac{\lambda^{3}}{v_{c}}\right)^2\]
03

Establish Relationship and Show Proportionality Constant

By choosing \(\beta_{1}\) and \(\beta_{2}\) such that\[\frac{1}{2}\beta_{1}\left(\frac{\lambda^{3}}{v_{c}}\right)=\frac{2}{3} \beta_{2}\left(\frac{\lambda^{3}}{v_{c}}\right)^2\]the left side can be set equal to 3, establishing the relationship\[\frac{P_{c} v_{c}}{k T_{c}}=3\]confirming that at the critical point \(k T_{c} / P_{c} v_{c} = 3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Irreducible Cluster Integrals
Understanding the behavior of gases involves various complexities, and the concept of irreducible cluster integrals plays a pivotal role in simplifying these complexities. In the virial expansion of the equation of state, irreducible cluster integrals, denoted as \( \beta_j \) for the jth cluster, are quantities that give us information about the interactions between groups of molecules.

Cluster integrals are termed 'irreducible' because they represent contributions from interacting groups of particles that cannot be decomposed into smaller independent clusters. In essence, these integrals are a measure of the probability that a cluster of molecules will form, considering the influence of interactions that cannot be captured by just looking at individual particles or pairs.

Specifically in our exercise, the virial expansion truncated after the second cluster integral interprets a gas where only one-body and two-body interactions are considered significant. This approximation is particularly useful around the critical point, where the behavior of gases is often non-ideal and complex.
Critical Region in Thermodynamics
The critical region in thermodynamics refers to the condition surrounding the critical point, an intriguing state where the properties of the gas and liquid phase converge. At the critical point, characterized by critical temperature \( T_c \), critical pressure \( P_c \), and critical volume \( v_c \), the distinction between the liquid and gaseous phase of a substance disappears.

In this unique state, the substance undergoes critical phenomena such as large fluctuations in density and the occurrence of an infinite correlation length. As such, gases do not obey the ideal gas law in this region, necessitating the use of more complex equations, such as the virial expansion used in our exercise, to describe their behavior. The key finding in the step-by-step solution is that specific relationships between the irreducible cluster integrals must hold true at the critical point, revealing the intricate balance of intermolecular forces at play.
Statistical Mechanics Equations
The laws of thermodynamics bridge into the microscopic world through statistical mechanics equations. These equations incorporate the principles of statistics and probability to relate the microscopic details of particles and their interactions to the macroscopic observables like temperature, pressure, and volume.

One such equation is the virial expansion, which expresses the deviation of a real gas from ideal behavior. It is built upon the notion that the pressure of a gas is not solely a product of the motion of non-interacting particles (as in an ideal gas) but also includes the effects of inter-particle forces. The virial coefficients in the expansion quantify these interactions.

Application in Real Gases

Real gases exhibit complex behaviors that cannot be fully described by the ideal gas law, especially near the critical point. Statistical mechanics provides the framework to derive equations like the virial expansion, which accounts for real gas behavior with increasingly accurate approximations by including more terms with higher-order cluster integrals.

In the context of our exercise, only the first two virial coefficients are significant in the critical region, indicating that statistical mechanics allows us to simplify the complex interactions in a real gas to a manageable number of terms around the critical point.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Liquid \(\mathrm{He}^{4}\) undergoes a superfluid transition at \(T \simeq 2.17 \mathrm{~K}\). The order parameter in this case is a complex number \(\Psi\), which is related to the Bose condensate density \(\rho_{0}\) as $$ \rho_{0} \sim|\Psi|^{2} \sim|t|^{2 \beta} \quad(t \leq 0) . $$ The superfluid density \(\rho_{s}\), on the other hand, behaves as $$ \rho_{s} \sim|t|^{v} \quad(t \leq 0) . $$ Show that the ratio \(^{23}\) $$ \left(\rho_{0} / \rho_{s}\right) \sim|t|^{\eta \nu} \quad(t \lesssim 0) $$

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 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 \((1960,1961)\).]

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 \(n\).

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
See all solutions

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Read Explanation

Space Physics

Read Explanation

Classical Mechanics

Read Explanation

Astrophysics

Read Explanation

Kinematics Physics

Read Explanation

Magnetism

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.