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Suppose you have 10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.

(a) Compute the average energy of all your atoms, by adding up all their energies and dividing by 10.

(b) Compute the probability that one of your atoms chosen at random would have energy E, for each of the four values of E that occur.

(c) Compute the average energy again, using the formulaE¯=sE(s)P(s)

Short Answer

Expert verified

Therefore,

The average energy is E¯=1.7eV

The probability is P(0eV)=410P(1eV)=310P(4eV)=210P(6eV)=110

The average energy using formula isE¯=1.7eV

Step by step solution

01

Given information

10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.

02

Explanation

(a) Assume we have ten weberium atoms, four of which have an energy of E1 = 0 eV, three of which have an energy of E2 =1 eV, two of which have an energy of E3 4 eV, and one of which has an energy of E4 = 6 eV; the average energy is given by:

E¯=NENN

Where ENis the energy level that occupied by N atoms

Substitute the values,

E¯=4(0eV)+3(1.0eV)+2(4eV)+1(6eV)10=1.7eVE¯=1.7eV

(b) Each energy's probability is equal to the number of atoms with that energy divided by the total number of atoms, so:

P(0eV)=410P(1eV)=310P(4eV)=210P(6eV)=110

03

Explanation

The average energy is:

E¯=E(s)P(s)

The average energy of the system is:

E¯=E1P(0eV)+E2P(1eV)+E3P(4eV)+E4P(6eV)

Substitute from part (b):

E¯=(0eV)(0.4)+(1eV)(0.3)+(4eV)(0.2)+(6eV)(0.1)E¯=1.7eV

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Most popular questions from this chapter

For a CO molecule, the constant ϵis approximately 0.00024eV. (This number is measured using microwave spectroscopy, that is, by measuring the microwave frequencies needed to excite the molecules into higher rotational states.) Calculate the rotational partition function for a COmolecule at room temperature (300K), first using the exact formula 6.30 and then using the approximate formula 6.31.

This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. As in Section 2.2, the allowed energies of each oscillator are 0, hf, 2hf, and so on. (

a) Prove by long division that

11-x=1+x+x2+x3+

For what values of x does this series have a finite sum?

(b) Evaluate the partition function for a single harmonic oscillator. Use the result of part (a) to simplify your answer as much as possible.

(c) Use formula 6.25 to find an expression for the average energy of a single oscillator at temperature T. Simplify your answer as much as possible.

(d) What is the total energy of the system of N oscillators at temperature T? Your result should agree with what you found in Problem 3.25.

(e) If you haven't already done so in Problem 3.25, compute the heat capacity of this system and check t hat it has the expected limits as T0 and T.

Carefully plot the Maxwell speed distribution for nitrogen molecules at T=300K and atT=600K. Plot both graphs on the same axes, and label the axes with numbers.

Calculate the most probable speed, average speed and rms speed for OxygenO2molecules at room temperature.

2. Consider a classical particle moving in a one-dimensional potential well u(x), as shown The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics.

{a) Show that the average position of the particle is given by

where each integral is over the entire xaxis.

A one-dimensional potential well. The higher the temperature, the farther the particle will stray from the equilibrium point.

(b) If the temperature is reasonably low (but still high enough for classical mechanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand u(x)in a Taylor series about the equilibrium point x0: u(x)=ux0+x-x0dudxx0+12x-x02d2udx2x0

+13!x-x03d3udx3x0+

Show that the linear term must be zero, and that truncating the series after the quadratic term results in the trivial prediction x=x0.

(c) If we keep the cubic term in the Taylor series as well, the integrals in the formula for xbecome difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from zo by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series foru(x)

(d) The interaction of noble gas atoms can be modeled using the Lennard Jones potential,

u(x)=u0x0x12-2x0x6

Sketch this function, and show that the minimum of the potential well is at x=x0, with depth u0. For argon, x0=3.9Aand u0=0.010eV. Expand the Lennard-Jones potential in a Taylor series about the equilibrium point, and use the result of part ( c) to predict the linear thermal expansion coefficient of a noble gas crystal in terms of u0. Evaluate the result numerically for argon, and compare to the measured value

α=0.0007K-1(at80K)

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