Answer ALL THREE questions.
The numbers in square brackets show the provisional
allocation of maximum marks per question or part of
question.
[Part marks]
1. Consider a container of volume V that contains N non-interacting
particles and that is in thermal equilibrium with its environment at
a temperature T, where you may assume that N 1, and hence
the Stirling formula ln N! = N ln N − N. The partition function of a
single particle (with zero spin) is given by
Z1 =
V
λ
3
th
.
(a) What does λth refer to in this expression? In a few words,
[2]
state the consequences of the average inter-particle distance
being λth and of it being . λth.
(b) Use the Gibbs entropy S and the canonical distribution to
[3]
show that
S =
1
T
h Ei + kB ln Z ,
where Z refers to the partition function of the system and h Ei
is the mean energy of the system.
(c) Using the results above, determine the entropy Sdist and
[3]
Sindist in terms of h Ei , N, V and λth, for the cases of the
particles being distinguishable and indistinguishable,
respectively. Explain your answer.
(d) Show that Sindist is an extensive variable and that Sdist is not.
[5]
Explain why this is so.
This question continues on the next page.
PHAS0024/2020 CONTINUED
1[Part marks]
(e) Two such containers — in thermal equilibrium at temperature
T and containing many non-interacting particles — are now
put into contact, such that they can exchange particles. For
convenience, we label their volumes V1 and V2, and the
numbers of indistinguishable particles in them N1 and N2,
respectively, with N1 + N2 = N = constant. We keep track of
how many particles are in V1 and how many are in V2.
i. Taking into account the number of possible permutations
[3]
of the particles, provide an expression for the partition
function ZV1+V2 of this combined system.
ii. By appropriate maximisation of SV1+V2 , show that the
[4]
particles are distributed such that N1/N2 = V1/V2. You
may assume that N1, N2 1.
PHAS0024/2020 CONTINUED
2[Part marks]
2. (a) A (single) protein binds from a reservoir at temperature T and
chemical potential µ to a (single) binding site. The unbound
and bound states have energies 0 and b < 0, respectively.
i. What is the grand canonical partition function ZG for this
[2]
system?
ii. Show that, in general, the average number of particles
[3]
h Ni can be written as a partial derivative of ln ZG.
iii. Show that
[3]
h Ni =
1
e
β( b−µ) + 1
(b) As a toy model for dimerisation of proteins on a membrane,
consider a canonical ensemble of two proteins bound to the
sites of a square lattice. The two proteins are a dimer if they
occupy adjacent (nearest neighbour) sites on the lattice, and
are monomers otherwise. The energies of the monomeric
and dimeric states are 0 and D, respectively. The lattice
contains L × L = M sites, where you may assume that L 1
and that the multiplicity of the dimeric state is small compared
to that of the monomeric state.
i. What is the multiplicity of the monomeric state for these
[3]
two proteins on the lattice? Explain your answer.
ii. What is the multiplicity of the dimeric state for these two
[3]
proteins on the lattice? Explain your answer.
iii. Derive expressions for the Helmholtz free energy of the
[4]
monomeric state and of the dimeric state. Show that the
two proteins are more likely to form a dimer than to be
monomers if
D < −kBT ln M
4 ≡ ∗ .
iv. Explain why | ∗ | increases with T and with M.
[2]
PHAS0024/2020 CONTINUED
3[Part marks]
3. (a) To describe the behaviour of conduction electrons in a metal,
we often consider these electrons as an ideal gas, and we
can use ideal gas results to calculate the heat capacity.
i. State the equipartition theorem.
[2]
ii. Using the equipartition theorem, show that the heat
[3]
capacity of a classical ideal gas of N particles is given by
C =
3
2
k
BN .
(b) For a metal, however, we fifind that the heat capacity due to
conduction electrons is given by
C ∝ kBN
T
TF
.
i. What does TF refer to in this expression? What is the
[2]
physical meaning of the energy that it corresponds to?
ii. Explain which specifific property of electrons defifines this
[2]
behaviour and is responsible for the deviation from the
prediction for a classical ideal gas.
This question continues on the next page.
PHAS0024/2020 CONTINUED
4(c) In a superconductor at suffificiently low temperature, electrons
form pairs that occupy a ground state with an energy defifined
as = 0. The lowest-energy excited state can only be
accessed by breaking these pairs, at a cost of ∆ per
electron. We here consider the limit kBT ∆, such that we
can ignore all higher excited states and can assume that
electrons only occupy states with energies 0 and ∆.
i. Explain in a few words why electron pair formation is a
[2]
requirement for superconductivity to occur.
ii. Show that in this limit (kBT ∆), the heat capacity of a
[6]
superconductor due to these electrons obeys
C ∝ ∆
T 2
exp − k ∆
BT .
iii. In this limit, compare the heat capacity of a
[3]
superconductor with that of a metal, sketching them
together in an annotated plot.
PHAS0024/2020 END OF EXAMINATION PAPER
担心学业？你还有其他选择！
KJEssay 学年守护计划！
我们是全网首家积极根据新政策优化应对方案的论文服务机构！
全面升级给你最好的防护！
1、远程代劳，资料下载，作业提交，有需要全程代劳！
KJEssay已对目前主流的教学系统Blackboard、ReCap，以及各校的ePortfolio，对全体老师做过专项培训，这方面有困难的学生，可直接授意老师代劳，我们将为你全面服务！
2、考核考试，老师提前充分备考，同程协助，助力满分！
KJEssay 保障学业提供全面服务！专业老师团队先学习了解课程内容，做充足应对，设计方案，
考试时，老师，专业应急团队，客服，同时待命！
老师快速反应，迅速做出最佳答案以及思路！
应急团队集思广益可对重难点迅速突破！
客服居中，全面负责协调沟通，提高效率！
给予及时而效率的全面帮助！
3、远程上课，录屏打卡课程讨论一个不落！
针对目前在线网课，KJEssay做出专项研究，对包括Autodesk、Azure、Skype、Zoom等视频教学软件有着充分熟悉。上网打卡一个不落。
4、保障隐私安全，全程一人全面追踪服务！所有人均签有隐私合同！
全面服务将主要安排在一位老师全面负责，做好对信息情况的充足了解掌握，不假他手！更因为全程彻底的参与，对情况以及考试有更彻底的把握！更能依据情况做出应对！也更易获取更高分！
客服以及第三方，时刻追踪，定期反馈情况。
5、一举一动全面反馈！时刻监控，看得到的全过程！24小时客服待命！
我们一直把沟通反馈，放在重中之重！尤其是代理服务，最了解的肯定还是客户，所以KJEssay会反馈所有的情况，没有客户允许下，不擅专！不乱动！
以最安全的形式，保障拿到最好的成绩！
在上半年的全面代理中,现已取得了优异的成绩与效果。
新学期，我们应对留学网课，更有经验，更加从容！
关于KJEssay
我们是KJEssay，31639人的选择！
现在就可联系我们
微信->添加朋友->添加企业微信联系人：13262280223
官网：https://www.kjessay.com
邮箱：kaijiewrite@163.com service@kjessay.com
WhatsApp：+44 7410496844（推荐添加）
QQ：1483266981
立即联系我们参与活动吧~