Modern Physics
2nd Edition
ISBN: 9780805303087
Author: Randy Harris
Publisher: Addison Wesley
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 5, Problem 54E
(a)
To determine
Minimum potential energy and the separation
(b)
To determine
The spring constant.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
In the simple kinetic theory of a gas we discussed in class, the molecules are assumed to be point-like objects (without any volume) so that they rarely collide with one another. In reality, each molecule has a small volume and so there are collisions. Let's assume that a molecule is a hard sphere of radius r. Then the molecules will occasionally collide with each other. The average distance traveled between two successive collisions (called mean free path) is λ = V/(4π √2 r2N) where V is the volume of the gas containing N molecules. Calculate the mean free path of a H2 molecule in a hydrogen gas tank at STP. Assume the molecular radius to be 10-10
a) 2.1*10-7 m
b) 4.2*10-7 m
c) none of these.
Problem 1:
This problem concerns a collection of N identical harmonic oscillators (perhaps an
Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf,
and so on.
a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving
1-x
the formula by long division. Prove it by first multiplying both sides of the
equation by (1 – x), and then thinking about the right-hand side of the resulting
expression.
b) Evaluate the partition function for a single harmonic oscillator. Use the result of
(a) to simplify your answer as much as possible.
c) Use E = -
дz
to find an expression for the average energy of a single oscillator.
z aB
Simplify as much as possible.
d) What is the total energy of the system of N oscillators at temperature T?
One description of the potential energy of a diatomic molecule is given by the Lennard–Jones potential, U = (A)/(r12) - (B)/(r6)where A and B are constants and r is the separation distance between the atoms. Find, in terms of A and B, (a) the value r0 at which the energy is a minimum and (b) the energy E required to break up a diatomic molecule.
Chapter 5 Solutions
Modern Physics
Ch. 5 - Prob. 1CQCh. 5 - Prob. 2CQCh. 5 - Prob. 3CQCh. 5 - Prob. 4CQCh. 5 - Prob. 5CQCh. 5 - Prob. 6CQCh. 5 - Prob. 7CQCh. 5 - Prob. 8CQCh. 5 - Prob. 9CQCh. 5 - Prob. 10CQ
Ch. 5 - Prob. 11CQCh. 5 - Prob. 12CQCh. 5 - Prob. 13CQCh. 5 - Prob. 14CQCh. 5 - Prob. 15CQCh. 5 - Prob. 16CQCh. 5 - Prob. 17CQCh. 5 - Prob. 18CQCh. 5 - Prob. 19ECh. 5 - Prob. 20ECh. 5 - Prob. 21ECh. 5 - Prob. 22ECh. 5 - Prob. 23ECh. 5 - Prob. 24ECh. 5 - Prob. 25ECh. 5 - Prob. 26ECh. 5 - Prob. 27ECh. 5 - Prob. 28ECh. 5 - Prob. 29ECh. 5 - Prob. 30ECh. 5 - Prob. 31ECh. 5 - Prob. 32ECh. 5 - Prob. 33ECh. 5 - Prob. 34ECh. 5 - Prob. 35ECh. 5 - Prob. 36ECh. 5 - Prob. 37ECh. 5 - Prob. 38ECh. 5 - Prob. 39ECh. 5 - Prob. 40ECh. 5 - Prob. 41ECh. 5 - Prob. 42ECh. 5 - Obtain expression (5-23) from equation (5-22)....Ch. 5 - Prob. 44ECh. 5 - Prob. 45ECh. 5 - Prob. 46ECh. 5 - Prob. 47ECh. 5 - Prob. 48ECh. 5 - Prob. 49ECh. 5 - Prob. 50ECh. 5 - Prob. 51ECh. 5 - Prob. 52ECh. 5 - Prob. 53ECh. 5 - Prob. 54ECh. 5 - Prob. 55ECh. 5 - Prob. 56ECh. 5 - Prob. 57ECh. 5 - Prob. 58ECh. 5 - Prob. 59ECh. 5 - Prob. 60ECh. 5 - Prob. 61ECh. 5 - Prob. 62ECh. 5 - Prob. 63ECh. 5 - Prob. 64ECh. 5 - Prob. 65ECh. 5 - Prob. 66ECh. 5 - Prob. 67ECh. 5 - Prob. 68ECh. 5 - Prob. 69ECh. 5 - Prob. 70ECh. 5 - Prob. 71ECh. 5 - In a study of heat transfer, we find that for a...Ch. 5 - Prob. 73CECh. 5 - Prob. 74CECh. 5 - Prob. 75CECh. 5 - Prob. 76CECh. 5 - Prob. 77CECh. 5 - Prob. 78CECh. 5 - Prob. 79CECh. 5 - Prob. 80CECh. 5 - Prob. 81CECh. 5 - Prob. 82CECh. 5 - Prob. 83CECh. 5 - Prob. 84CECh. 5 - Prob. 85CECh. 5 - Prob. 86CECh. 5 - Prob. 87CECh. 5 - Prob. 88CECh. 5 - Consider the differential equation...Ch. 5 - Prob. 90CECh. 5 - Prob. 91CECh. 5 - Prob. 92CECh. 5 - Prob. 93CECh. 5 - Prob. 94CECh. 5 - Prob. 95CECh. 5 - Prob. 96CECh. 5 - Prob. 97CECh. 5 - Prob. 98CE
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- Assume that air is an ideal gas under a uniform gravitational field, so that the potential energy of a molecule of mass m at altitude z is mgz. Show that the distribution of molecules varies with altitude as given by the distribution function f(z) dz = Cz exp(-βmgz) dz and that the normalization constant Cz= mg/kT. This distribution is referred to as the law of atmospheres.arrow_forwardConsider a mass, m, moving under the influence of an effective potential energy -a b T 7-2 U(r) = + " where a and b are positive constants and r is the radial distance from the origin. In this case, U(r) is a 1D potential energy. (c) Expand The function for U(r) for small displacements about the equilibrium point, To. This will be a Taylor series for U(r) in terms of (r-ro)", where n is an integer. Generate the first three nonzero terms in the series.arrow_forwardUsing the same procedure to determine the fundamental equation of chemical thermodynamics (dG = –SdT + VdP) from the Gibbs free energy of a system (G = H – TS), can you please explain how to find the analogous fundamental equation for (A=U-TS)? Also, can you please handwrite the formula down instead of typing? I get confused with typed formulas sometime.arrow_forward
- Consider two immiscible liquids such as water and oil. If a spherical oil molecule of radius r is taken out of the oil phase and placed in the water phase, the unfavorable energy of this transfer is proportional to the area of the solute (oil) molecule newly exposed to the solvent (water) multiplied by the interfacial energy, i, of the oil-water interface. The interfacial energy of the bulk cyclohexane-water interface is i = 50 mJ m-2, and the radius of a cyclohexane molecule is 0.28 nm. Using Boltzmann distribution, estimate the solubility of cyclohexane in water at 25 C in units of mol L-1.The concentration of water in water phase is 55.5 mol L-1.arrow_forwardUsing MATLAB editor, make a script m-file which includes a header block and comments: Utilizing the ideal gas law: Vmol= RT/P Calculate the molecular volume where: R = 0.08206 L-atm/(mol-K) P = 1.015 atm. and T = 270 - 315 K in 5 degree increments Make a display matrix which has the values of T in the first column and Vmol in the second column Save the script and publish function to create a pdf file from the script in a file named "ECE105_Wk2_L1_Prep_1"arrow_forwardInterstellar space is quite different from the gaseous environments we commonly encounter on Earth. For instance, a typical density of the medium is about 1 atom cm−3 and that atom is typically H; the effective temperature due to stellar background radiation is about 10 kK. Estimate the diffusion coefficient and thermal conductivity of H under these conditions. Compare your answers with the values for gases under typical terrestrial conditions. Comment: Energy is in fact transferred much more effectively by radiation.arrow_forward
- A molecule has states with the following energies: 0, 1ε, 2ε, 3ε, and 4ε, where ε = 1.0 x 10-20 J. Calculate the probability that a molecule is in the ground state (with zero energy) for a collection of molecules in thermal equilibrium at T = 300 K. Provide your answer as a number in normal form to 3 decimal places (in the form X.XXX). It is a good idea to keep 4 decimal places during your calculation, then round to 3 decimal places for your submitted answer. Hint: note that this molecule has a finite number of states so you must take a finite sum, do not use expressions for infinite sums. Also note that your calculations for this problem will be useful for the next two problems, so keep them.arrow_forwardThe Clausius-Clapeyron relation 5.47 is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and ~V depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take ~V to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:This result is called the vapor pressure equation. Caution: Be sure to use this formula only when all the assumptions just listed are valid.arrow_forwardGlasses are materials that are disordered – and not crystalline – at low temperatures. Here is a simple model. Consider a system with energy E, the number of accessible microstates is given by a Gaussian function: Ω(E) = Ω0 e −(E−E¯) 2/(2∆2 ) , where Ω0 and E¯ and ∆ are positive constants. E¯ is the average energy. In this problem, we consider only the states whose energy E is below E¯. 1. Show that the entropy is an inverted parabola: S(E) = S0 − α (E − E¯) Find S0 and α. Write your answers in terms Ω0, ∆, and other universal constants. 2. An entropy catastrophy happens when S = 0, which occurs at energy E0. (i) Find E0. (ii) What is the number of accessible states for energy below E0? 3. The glass transition temperature Tg is the temperature of entropy catastrophy. Compute Tg. 4. Find the energy E as a function of T. 5. Without any calculation, explain why E → E¯ as T → +∞. Hint: consider the Boltzmann distributionarrow_forward
- Problem 1: In statistical mechanics, the internal energy of an ideal gas is given by: N. aNkB 2/3 (3NKB U = U(S,V) = е where a is a constant. 1- Show that the variation of the internal energy is given by: 2 dS - \3V 2 dU = dV \3NkB 2- Using the fundamental relation of thermodynamic dU = T.ds – p. dV, show that the equation of state PV = nRT follows from the first expression of U.arrow_forwardIn the Taylor Expansion, it looks like they've made x_0 = x_1, x_2 and made x = x_1 - a, x_2 - a. I don't understand how they can use that substitution for x_0 since it is a constant, and x_1 and x_2 are variables that can change.arrow_forwardConsider N identical harmonic oscillators (as in the Einstein floor). Permissible Energies of each oscillator (E = n h f (n = 0, 1, 2 ...)) 0, hf, 2hf and so on. A) Calculating the selection function of a single harmonic oscillator. What is the division of N oscillators? B) Obtain the average energy of N oscillators at temperature T from the partition function. C) Calculate this capacity and T-> 0 and At T-> infinity limits, what will the heat capacity be? Are these results consistent with the experiment? Why? What is the correct theory about this? D) Find the Helmholtz free energy from this system. E) Derive the expression that gives the entropy of this system for the temperature.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning