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Chapter 7: Q. 7.68 (page 323)

Calculate the condensate temperature for liquid helium-4, pretending that liquid is a gas of noninteracting atoms. Compare to the observed temperature of the superfluid transition, 2.17K. ( the density of liquid helium-4 is 0.145g/cm3)

Short Answer

Expert verified

The condensate temperature for liquid helium-4 is 3.13K

Step by step solution

01

Step 1. Given information

Condensate temperature for a given substance =

Tc=1k0.527h22πmNV23

massofparticle,m=4(1.66×10-27)kg

For density of H4eliquid,

mHeV=0.145g/cm3

Now,

NV=mHeVmHeN

Putting the values of mHeV=0.145g/cm3andmHeN=4.00g/mol.

NV=0.145×103g/cm34.00g/mol

=0.036mol/cm3

=0.036mol/cm36.022×1023atoms1mole106cm31m3

=2.18×1028atoms/m3

02

Step 2. Putting the values of h, N/V, k, m in equation Tc=1k0.527h22πmNV23

we get,

Te=11.381×10-23J/K(0.527)6.63×10-34J·s22π(4)1.66×10-27kg2.18×1028atoms/m323

=3.13K

Therefore the condensate temperature for liquid helium-4 is 3.13K.

This value is similar to experimental value of 2.17K, neglecting all the interaction between Heatoms

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Most popular questions from this chapter

Consider a free Fermi gas in two dimensions, confined to a square area A=L2•

(a) Find the Fermi energy (in terms of Nand A), and show that the average energy of the particles is ∈F2.

(b) Derive a formula for the density of states. You should find that it is a constant, independent of ∈.

(c) Explain how the chemical potential of this system should behave as a function of temperature, both when role="math" localid="1650186338941" kT≪∈Fand when Tis much higher.

(d) Because g∈is a constant for this system, it is possible to carry out the integral 7.53 for the number of particles analytically. Do so, and solve for μas a function of N. Show that the resulting formula has the expected qualitative behavior.

(e) Show that in the high-temperature limit, kT≫∈F, the chemical potential of this system is the same as that of an ordinary ideal gas.

Sketch the heat capacity of copper as a function of temperature from 0to5K, showing the contributions of lattice vibrations and conduction electrons separately. At what temperature are these two contributions equal?

The heat capacity of liquid H4ebelow 0.6Kis proportional to T3, with the measured valueCV/Nk=(T/4.67K)3. This behavior suggests that the dominant excitations at low temperature are long-wavelength photons. The only important difference between photons in a liquid and photons in a solid is that a liquid cannot transmit transversely polarized waves-sound waves must be longitudinal. The speed of sound in liquid He4is 238m/s, and the density is 0.145g/cm3. From these numbers, calculate the photon contribution to the heat capacity ofHe4in the low-temperature limit, and compare to the measured value.

Consider a gas of noninteracting spin-0 bosons at high temperatures, when T≫Tc. (Note that “high” in this sense can still mean below 1 K.)

  1. Show that, in this limit, the Bose-Einstein function can be written approximately as
    n¯BE=e−(∈−μ)/kT[1+e−∈−μ/kT+⋯].
  2. Keeping only the terms shown above, plug this result into equation 7.122 to derive the first quantum correction to the chemical potential for gas of bosons.
  3. Use the properties of the grand free energy (Problems 5.23 and 7.7) to show that the pressure of any system is given by In P=(kT/V), where Zis the grand partition function. Argue that, for gas of noninteracting particles, In Zcan be computed as the sum over all modes (or single-particle states) of In Zi, where Zi; is the grand partition function for the ithmode.
  4. Continuing with the result of part (c), write the sum over modes as an integral over energy, using the density of states. Evaluate this integral explicitly for gas of noninteracting bosons in the high-temperature limit, using the result of part (b) for the chemical potential and expanding the logarithm as appropriate. When the smoke clears, you should find
    p=NkTV(1−NvQ42V),
    again neglecting higher-order terms. Thus, quantum statistics results in a lowering of the pressure of a boson gas, as one might expect.
  5. Write the result of part (d) in the form of the virial expansion introduced in Problem 1.17, and read off the second virial coefficient, B(T). Plot the predicted B(T)for a hypothetical gas of noninteracting helium-4 atoms.
  6. Repeat this entire problem for gas of spin-1/2 fermions. (Very few modifications are necessary.) Discuss the results, and plot the predicted virial coefficient for a hypothetical gas of noninteracting helium-3 atoms.

Consider two single-particle states, A and B, in a system of fermions, where ϵA=μ-xand ϵB=μ+x; that is, level A lies below μ by the same amount that level B lies above μ. Prove that the probability of level B being occupied is the same as the probability of level A being unoccupied. In other words, the Fermi-Dirac distribution is "symmetrical" about the point where ϵ=μ.
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