BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANISECOND SEMESTER 2014-15BITS F111: Thermodynamics

Problem set 3

Problem Solving Session (1/02/2015)

Instructions

1. Consult the Instructor in case you need any clarifications.

1 Scale height

In the atmospheric pressure law

n = n0e− mgh

kBT

kBT/mg = RT/Mg = h0 is called the scale height ( M is the molecular weight). Evaluate the scale heightfor Earth’s atmosphere and Sun’s atmosphere, given ME = 29, TE = 300K, MS = 1.5 and TS = 5500K.

2 Dalton’s law of partial pressures

A volume V at temperature T contains nA and nB moles of ideal gases A and B respectively. The gases donot react chemically. Show that the total pressure is given by

P = PA + PB

where PA and PB are the pressures that each gas would exert if it were in the volume alone.

In diatomic gases some molecules are dissociated into separated atoms, the fraction dissociated increasing withtemperature. On the whole at a given temperature the gas consistes of a monoatomic and diatomic portion.The mixture does not behave like an ideal gas though each component separately does. If m be the total massof the gas and m1 be the mass of the monoatomic portion, the degree of dissociation is defined as δ = m1/m.What is the equation of state of the gas. The molecular weight of the diatomic component may be taken asM2.

3 Virial expansion

The following expansion is called Virial expansion

Pv = RT (1 +BP + CP 2 + · · · )

Determine the second Virial coeffecient B in each of the following cases1. (

P +a

v2

)(v − b) = RT

2. (Pea/RTv

)(v − b) = RT

3. (P +

a

v2T

)(v − b) = RT

4 Compressibility and expansivity

1. A hypothetical substance has isothermal compressibility κ = a/v and expansivity β = 2bT/v, where a andb are constants. At a pressure of P0 and temperature of T0 the specific volume is v0. Find the equation ofstate.

2. A substance has an isothermal compressibility κ = aT 3/P 2 and expansivity β = bT 2/P , where a and bare constants. Find the equation of state.

3. Using the fact that dv is an exact differential show that(∂β

∂P

)T

= −(∂κ

∂T

)P

4. The equilibrium states of superheated steam is represented by Callendar’s equation

1

2 PDF LATEX coloured text and graphics

v − b =rT

P− a

Tm

where b, r, a and m are constants. Calculate the volume expansivity β as a function of T and P .

5. Consider a wire with cross section area A that undergoes linear expansion from an initial equilibrium to afinal equilibrium state. Show that the change in the tension is

dF = −αAY dT +AY

LdL

where α is coeffecient of linear expasion and Y is the Young’s modulus. For an equation of state of the form

FKT

=L

L0− L0

2

L2

find the isothermal Young’s modulus.

5 Poisson Distribution in Molecules of a gas. Continued fromthe first problem set

The Stirling approximation used in the problem on the distribution for fluctuations for large N is givenby

ln(N !) ∼ N ln(N)−Nwhich maybe written as

N ! ∼ (N/e)N

Upto the leading order we have a logarithmic correction

ln(N !) ∼ N lnN −N +O(ln N)

The O(ln N) term was earlier ignored as it clearly is much smaller than the dominant contribution from thefirst two terms. This correction is actually given by ln

√(2πN) which yields the following

N ! ∼√

2πN

(N

e

)NLet us now do a more general problem. Calculate the probability of having n particles in a small volume V ofa bigger box with total volume κV and a total number of particles κN0. For κ = 2 we should get the earlierresult. Show that the probability maybe written in the form

Pn =ane−a

n!

Identify a in this case. This is infact a well known probability distribution called the Poisson distribution.Show that you may get the Gaussian approximation from this and reproduce the earlier result for κ = 2.

6 Work

1. Consider a quasistatic adiabatic expansion of a gas contained in a vessel when the pressure at any momentis given by

PV γ = K,

where K and γ are constants. Calculate the work done when the expansion takes place from an initial state(Pi, Vi) and the final state is (Pf , Vf ).

2. Calculate the work done upon expansion of 1 mole of a Van der Waal gas isothermally from volume vi tovf .

3. The tension in a wire is increased quasi statically and isothermally from Fi to Ff . The length L area Aand isothermal Young’s modulus remains practically constant. Calculate the work done.

4. For the equation of state given in problem 4.5, calculate the isothermal work done in compressing thelength from L = L0 to L = L0/2.

5. Consider a substance where the internal energy function is given by

U = 2.5PV + constant

Find the equation of adiabats in the P − V plane. What would be the equation of the adiabats if the energyfunction was instead given by

U = aP 2V