Download Department of Chemistry, IIT-Delhi CY110N Tutorial

Department of Chemistry, IIT-Delhi
CY110N Tutorial Sheet 8: Second Law of Thermodynamics
Instructor: C. Chakravarty
1. Calculate the changes in entropy when a perfect gas is compressed to half its volume and
simulataneously heated to twice its initial temperature.
2. Two identical perfect gases with the same pressure P and the same number of particles N ,
but with different temperatures T1 and T2 are confined to two vessels of volume V1 and V2 .
What is the change in entropy when the two vessels are connected?
3. The standard reaction Gibbs energy of
K4 [Fe(CN)6 ]·3H2 O→ 4K+ (aq) + [Fe(CN)6 ]4− (aq) +3H2 O(l)
is 26.12 kJ/mol. The enthalpy of solution of the trihydrate is +55.0 kJ/mol. Calculate the
standard molar entropy of solution of the hexacyanoferrate (II) ion in water given that the
standard molar entropy of the solid trihydrate is 599.7 JK−1 mol−1 , that of the K+ ion in
water is 102.5 JK−1 mol−1 and that of H2 O(l) is 69.91 J/K mol.
4. The constant volume molar heat capacity of a solid may be approximated by Cv =
12 4
π R
5
!3
T
θD
if T << θD . The constant θD is different for each solid. For Al, θD =398 K. Find a formula
for the entropy change on heating a solid from temperature Ti to Tf .
5. Compare the difference between ∆H and ∆U for the following processes: (i) 1 mol H2 O(s)
→ 1 mol H2 O(l) at 273 K [Vm (ice)=0.0196 L.mol−1 , Vm (liq. water)=0.018 L.mol−1 ] (ii) 1
mol H2 O(l) → 1 mol H2 O(g) at 373 K [Vm (ice)=0.0188 L.mol−1, Vm (gaseous water)=30.61
L.mol−1 ] (iii) Combustion of α D-glucose (iv) 1 mol C(graphite) → 1 mol C (diamond) at a
pressure of 20,000 atm
6. Which
of these
cyclic integrals must
vanishH for a closed
system with
P -V work only? (a)
H
H
H
H
H
P dV ; (b) (P dV + V dP ); (c) V dV ; (d) HdT ; (e) drev ; (f) dwrev /P .
7. The purest acetic acid is often called glacial acetic acid because it is purifoed by fractional
freezing at its melting point of 16.6C. A flask containing several moles of acetic acid is lowered
into an ice-water bath briefly. When it is removed, it is found that exactly 1 mol of acetic
acid has frozen. Given that ∆Hf us for water and acetic acid are 5.98 kJ/mol and 11.45
kJ/mol respectively, find: (a) the change in entropy of the acetic acid (b) change in entropy
of the water bath (c) Now consider that the water bath and the acetic acid are in the same
system, What is the entropy change for the combined system? Is the process reversible or
irreversible? Why?
8. One mol of hydrogen occupies a volume of 0.1 m3 at 300K and one mol of argon also occupies
the same volume but at 400K. While isolated from their surroundings, each undergoes a free
expansion, the hydrogen to 5 times and argon 8 times its initial volume. The two masses
are then placed in contact with each other and reach thermal equilibrium. What is the total
Ar
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change in entropy? (CH
v = 10 kJ/K/kg, Cv = 0.31 kJ/K/kg)
9. One mole of an ideal monoatomic gas undergoes an irreversible adiabatic process in which
the gas ends up at STP and for which ∆S is 21JK−1 and w is 1.26 kJ. The entropy of the
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gas at STP is 270 JK−1 mol−1 . Calculate ∆U and ∆G for the process and also the initial
state of the gas.
10. Calculate the ratio of the maximum inversion temperature to the critical temperature for a
van der Waals gas.
11. Show that the Joule-Thomson coefficient for a real gas is not zero in the limit of zero pressure.
12. For a gas following the equation of state Vm =
RT
P
a
+ b − RT
!
what is the inversion tempera-
ture for the Joule-Thompson effect? What is the sign of µJT below the inversion temperature?
13. Calculate Cp − Cv for a van der Waals gas, keeping only first-order corrections to ideal gas
behaviour.
14. Prove the following two relations:
(a)
! ∂S
∂p
V
!
∂T
∂V
−
p
! ∂T
∂p
V
!
∂S
∂V
p
(b) µJ CV = p − (αT /κT ) !
∂p
T dS = Cv dT + T ∂T
(c)
dV
V
15. It is possible to cool liquid water below its freezing point of 273.15 K without the formation
of ice if proper care is taken to prevent nucleation. A kilogram of subcooled liquid water at
263.15 K is contained in a well-insulated vessel. Nucleation is induced by the introduction
of a speck of dust, and a spontaneous crystallization process ensues. Find the final state of
the water and calculate the total entropy change for the process.
16. 100 g of ice at 0 o C is dropped into an insulated beaker containing 150 g of water at 100 o C.
Calculate ∆S for this process.
17. Calculate the maximum work and the maximum non-expansion work that can be obtained
from the freezing of supercooled water at −5 o C and 1.0 atm. The densities of water and ice
are 0.999 and 0.917 g cm−3 , respectively at −5 o C.
18. One mole of He is heated from 200 o C to 400 o C at a constant pressure of 1 atm. Given that
the absolute entropy of He at 200 o C is 810 JK−1 mol−1 , and assuming He is a perfect gas,
comment on the spontaneity of the process.
19. (a) If S is regarded as a function of p and T , then show that T dS = Cp dT − αV T dp.
(b) Assume that when pressure is applied on a condensed phase, only the compressibility
κ remains constant but not the volume. This implies that to first order changes in volume
with pressure are given by V (p0 + ∆p) = V (p0 ) + κV (p0 )∆p where V (p0 ) is the volume at
some pressure p0 . Then find an expression for the energy transferred as heat when pressure
is increased isothermally and reversibly from Pi to Pf for a solid.
) = V − T ( ∂V
) .
20. (a) Show that ( ∂H
∂p T
∂T P
0
(b) Consider a gas with compression factor Z such that Z = 1+ BRTp where B 0 is a temperaturedependent virial coefficient. Find the expression for the change in the molar enthalpy, Hm ,
as pressure on the gas increases from p1 to p2 at constant temperature.
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