PHY 365: Thermal Physics

Homework: Sackur-Tetrode Relation

1. Argon is a monatomic ideal gas with molecular weight 39.95 g mol−1.

(a) Calculate the entropy of 1 mol of argon at 300 K and pressure 1bar.

(b) The gas is heated to 600 K at constant pressure. What is thechange in entropy?

(c) The gas is then expanded adiabatically until its volume is doubled.What is the change in entropy during this process?

2. Verify using the Sackur-Tetrode formula that the change in entropy ofa monatomic ideal gas between two states is zero if the relation

PfV5/3f = PiV

5/3i

is satisfied.

3. Consider an adiabatic process on a monoatomic gas and rederive (makesure to include explanations in your derivation) the relation

PfVγf = PiV

γi

from purely thermodynamic considerations. Try not to look at thederivation in chapter 1.

4. Show that during the quasistatic isothermal expansion of a monoatomicideal gas, the change in entropy is related to the heat input Q by thesimple formula

∆S =Q

T.

Later on, we’ll prove that this formula is valid for any quasi staticprocess.

5. In a certain system, the internal energy U is related to the particlenumber N , and volume V through

U = const ·N(N

V

)Dexp

[DS

NkB

]

1

(a) Show that the system satisfies the ideal gas law independent ofthe value of the constant D.

(b) Find the coefficient γ in the adiabatic equation of state PV γ =const and the heat capacities CP and CV of the system.

6. One can rewrite the Sackur-Tetrode formula under the form

S = NkB

(ln

[V/N

vq

]+

5

2

)where

vq = Λ3

Λ =

√h2

2πmkBT

Λ is called the thermal de Broglie wavelength (for a free ideal gas).(a) Verify this this equation is equivalent to the one written in yourtextbook.(b) What is the (significance) thermal de Broglie wavelength? Explain(c) What is the thermal de Broglie wavelength of Argon at room tem-perature?

7. Bonus: Assume that the entropy of a diatomic ideal gas is of the formof the Sackur-Tetrode formula, but with

vq = const · T−α

where the constant depends on molecular properties, and natural con-stants such as h, π, etc but not on the thermodynamic variablesV, T,N . What is the value of α if

γ =CPCV

= 1.4

2