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Chapter 5: Q17P (page 223)

Thebulk modulus of a substance is the ratio of a small decrease in pressure to the resulting fractional increase in volume:

B=-VdPdV.

Show thatB=(5/3)P, in the free electron gas model, and use your result in Problem 5.16(d) to estimate the bulk modulus of copper. Comment: The observed value is 13.4脳1010N/m2, but don鈥檛 expect perfect agreement鈥攁fter all, we鈥檙e neglecting all electron鈥搉ucleus and electron鈥揺lectron forces! Actually, it is rather surprising that this calculation comes as close as it does.

Short Answer

Expert verified

B=53(3.84脳1010N/m2)=6.4脳1010N/m2

Step by step solution

01

Definition of bulk modulus of a substance

The volumetric stress to volumetric strain ratio for any given material is known as the bulk modulus.

02

Showing that B = (5/3) P in the free electron gas model

We need to find bulk modulus in the free electron gas model. And after that, we need to estimate the bulk modulus of copper from the previous problem.

p=2Etot3V=23鈩�2kF55m蚁5/3

P=(3蟺2)2/3鈩�25m蚁5/3=AV-5/3,A=鈩�2(3蟺2)2/3(Nq)5/35m.

Bulk modulus is equal to:

role="math" localid="1658144523557" B=-VdPdV=-V-53AV-8/3=53AV-5/3鈥�鈥�鈥�鈥�=53PB=53P

Degeneracy pressure of copper was P=3.84脳1010鈥�Pa. So Bulk modulus, according to the formula is,For copper,

B=533.84脳1010鈥�N/m2=6.4脳1010鈥�N/m2.

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Most popular questions from this chapter

Find the average energy per free electron (Etot/Nd), as a fraction of the

Fermi energy. Answer:(3/5)EF

Imagine two noninteracting particles, each of mass m, in the infinite square well. If one is in the state蠄n(Equation 2.28 ), and the other in state 蠄1(l鈮�n), calculate localid="1658214464999" (x1-x2)2, assuming (a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions.

Suppose you could find a solution蠄(r1,r2,...,rz)to the Schr枚dinger equation (Equation 5.25), for the Hamiltonian in Equation 5.24. Describe how you would construct from it a completely symmetric function, and a completely anti symmetric function, which also satisfy the Schr枚dinger equation, with the same energy.

role="math" localid="1658219144812" H^=鈭�j=1Z-魔22m鈭�j2-14蟺o,0Ze2rj+1214蟺o,0鈭�j鈮�1Ze2rj-rk (5.24).

role="math" localid="1658219153183" H^蠄=E (5.25).

Imagine two non interacting particles, each of mass , in the one dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate 鉄�(x1-x2)2鉄�assuming
(a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state.)

The density of copper is8.96g/cm3,and its atomic weight is63.5g/mole

(a) Calculate the Fermi energy for copper (Equation 5.43). Assume d = 1, and give your answer in electron volts.

EF=魔22m3蚁蟺22/3 (5.43).

(b) What is the corresponding electron velocity? Hint: SetEF=1/2mv2Is it safe to assume the electrons in copper are nonrelativistic?

(c) At what temperature would the characteristic thermal energyrole="math" localid="1656065555994" (kBT,wherekBkBis the Boltzmann constant and T is the Kelvin temperature) equal the Fermi energy, for copper? Comment: This is called the Fermi temperature,TF

. As long as the actual temperature is substantially below the Fermi temperature, the material can be regarded as 鈥渃old,鈥� with most of the electrons in the lowest accessible state. Since the melting point of copper is 1356 K, solid copper is always cold.

(d) Calculate the degeneracy pressure (Equation 5.46) of copper, in the electron gas model.

P=23EtotV=23魔2kF510蟺2m=3蟺22/3魔25m蚁5/3
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