A. (a) Consider a canonical ensemble having N particle, V volume and at T temperature. Write down the expression of partition function (QON,V,T)) of this canonical ensemble in terms of the microstate energy E. (b) Write down the expression for Helmhotz free energy (A) and pressure (P) in terms of Q(N,V.T). (c) Now, assume that for a system of dense gas you can write down the Q(N,V,T) as, 1 2amk,T QUN,V.T) N! (V-Nb)" e Treat a and b as constants. Get the expression for pressure (P) in terms of V, a, b, N, kg and T. Rearrange that expression to get a form where in the RHS of the equation will have NkgT. Identify the equation.

A. (a) Consider a canonical ensemble having N particle, V volume and at T temperature. Write down the expression of partition function (QON,V,T)) of this canonical ensemble in terms of the microstate energy E. (b) Write down the expression for Helmhotz free energy (A) and pressure (P) in terms of Q(N,V.T). (c) Now, assume that for a system of dense gas you can write down the Q(N,V,T) as, 1 2amk,T QUN,V.T) N! (V-Nb)" e Treat a and b as constants. Get the expression for pressure (P) in terms of V, a, b, N, kg and T. Rearrange that expression to get a form where in the RHS of the equation will have NkgT. Identify the equation.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.36P

See similar textbooks

Similar questions

For an ideal gas of classical non- interacting atoms in thermal equilibrium, the Cartesian component of the velocity are statistically independent. In three dimensions, the probability density distribution of the velocity is: where σ² = kBT m P(Vx, Vy, Vz) = (2nо²)-³/² exp 20² 1. Show that the probability density of the velocity is normalized. 2. Find an expression of the arithmetic average of the speed. 3. Find and expression of the root-mean-square value of the speed. 4. Estimate the standard deviation of the speed.
Show that at high enough temperatures (where KBT » ħw) the partition function of a simple quantum mechanical harmonic oscillator is approximately Z≈ (Bħw)-¹ Then use the partition function to calculate the high temperature expressions for the internal energy U, the heat capacity Cy, the Helmholtz function F and the entropy S.
3) Four bones of eqyual sizes are to Contain distingulshable nine particles. Find the probability of macyostate j (3,401) and ii(3,3,3). the
Consider a canonical ensemble. (a) Show that: Ə In Z = kT² ƏT (b) Show that: z (aB2 (c) Recall the definition of heat capacity at constant volume: Cp = aT In the case of a canonical ensemble, is used for U in the above expression. Perform the necessary partial differentiation to obtain an expression for C, in terms of , , and kT. (d) Suppose we define: SE = E – Show that: kT² (e) Show that: -d - ²=
Consider a closed system with N noninteracting gaseous particles. Let the single particle nondegenerate energy levels be iЄ, i = 0,1,2,........ Evaluate the canonical partition function if the particles are (a) fermions (b) bosons. You are not expected to reach an answer here. This question is to illustrate the difficulty of using canonical ensembles.
Check Your Understanding If a=3+4i , what is the product a* a?
Check Your Understanding For the particle in the example, find the probability Of locating it 0 and L/4.
A particle with a size just below r* is unstable but a particle with a size 2) just above r* is stable. What comments would you make on this statement? If you agree then, why are the particles just above (very little bigger than r*) a critical size r* thermodynamically stable even though the energy (Gibbs free energy) is positive. a) A 4G, surface energy 4xr'y AG Volume free energy Excess free energy AG. b) Why the free energy vs particle radius curves for homogeneous and heterogeneous nucleation differs. CAG, AG AG JAGeter AGoma
Problem # 2. In the two-level system, estimate the emission line full width at half maximum (FWHM) for spontaneous emission at 650 nm if the spontaneous radiative lifetime of the upper state is about 3,000 nanoseconds.
Q-1 Calculate the minimum velocity at which spherical particles of density 1600 kg/m and of diameter 1.5 mm will be fluidised by water in a tube of diameter 10 mm on the assumption that the Carman-Kozeny equation is applicable. Discuss the uncertainties in this calcula- tion. Repeat the calculation using the Ergun equation and explain the differences in the results obtained.
At zero temperature, Nb3Sn has the critical fields Bel = = 26 mT and B2 = 29 T. Determine the coherence length (§), penetration depth (X), Ginzburg-Landau parameter (k), and thermodynamic critical field (B₂).
Calculate the ratio of spontaneous and stimulated emission light of frequency (5 x 10¹ HZ) at (2000 K°), and show that the equilibrium populations of levels energy of this transition?