×

# For a gas of particles confined inside a two-dimensional

## Problem 72P Chapter 7

An Introduction to Thermal Physics | 1st Edition

• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants

An Introduction to Thermal Physics | 1st Edition

4 5 0 337 Reviews
18
3
Problem 72P

For a gas of particles confined inside a two-dimensional box, the density of states is constant, independent of ϵ (see Problem). Investigate the behaviour of a gas of non interacting bosons in a two-dimensional box. You should find that the chemical potential remains significantly less than zero as long as T is significantly greater than zero, and hence that there is no abrupt condensation of particles into the ground state. Explain how you know that this is the case, and describe what does happen to this system as the temperature decreases. What property must g(ϵ) have in order for there to be an abrupt Bose-Einstein condensation?

Problem:

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

(a) Find the Fermi energy (in terms of N and A), and show that the average energy of the particles is ϵF/2

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

(c) Explain how the chemical potential of this system should behave as a function of temperature, both when. kT ≪ ϵF and when T is much higher.

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

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

Equation:

Step-by-Step Solution:
Step 1 of 3

March 8, 2016: CLASS NOTES CIRCULATION  ARTERIES/VEINS MAVE MUSCLES AROUND, MANY HAVE LARGER PROTECTIVE LAYERS  CAPILLARIES: SINGLE CELL LAYER WITH TONS OF PORES, LETS WATER AND SOLUTIONS FLOW ACCOUNTS FOR MASS MOVEMENT OF BLOOD THROUGH TISSUES LOOSE SPACES BETWEEN CELLS: CAN OPEN TO FIT WHOLE CELLS THROUGH BETWEEN ARTERIES (CAN’T CLOSE) AND VEINS  VASODILATION SPHINCTERS...

Step 2 of 3

Step 3 of 3

#### Related chapters

Unlock Textbook Solution

For a gas of particles confined inside a two-dimensional

×
Get Full Access to An Introduction To Thermal Physics - 1 Edition - Chapter 7 - Problem 72p

Get Full Access to An Introduction To Thermal Physics - 1 Edition - Chapter 7 - Problem 72p