1
Exam Thermodynamics
2 June 2016
Please, hand in your answers to problems 1, 2, 3 and 4 on separate sheets.
Put your name and student number on each sheet.
The examination time is 18:00 until 21:00.
There are 4 problems, with each 4 subproblems, a list of constants and a formulae sheet,
4 pages in total.
All 16 subproblems have equal weight for the final grade.
Problem 1
a) Give a definition and short description of the thermodynamic background for the following
concepts
– partial pressure
– reaction quotient
– ideal solution
– Boltzmann distribution
– work
b) Indicate for each of the following quantities whether we are dealing with a state function (all
symbols have their usual meaning); answer using only yes or no:
– V P,
– P V 2,
– PV − Q + TS − W,
– A,
– Qrev .
c) Give the meaning of all symbols in the following formula as well as a description of its use
in thermodynamic problems.
H − kT ln W
d) A system of molecules has a molecular vibrational energy spectrum given by i = i · , where
= 6 · 10−21 J, and i is a positive integer (i = 0, 1, 2, 3, 4, 5, .....).
At T = 300 K and P = P the probability for a molecule to occupy the vibrational state
labeled i = 7 equals 3.01 · 10−5 .
Calculate the probability for a molecule to occupy the vibrational state with energy 2 at
T = 300 K and P = P .
,2
Problem 2
In this problem we study the entropy change after an isochoric heating process for a perfect atomic
gas. We start the process with 2 mol of the gas in a closed cylinder with a volume V = 2 L, and
we heat the gas from 127◦ C to 177◦ C reversibly.
a) Determine the pressure at the start of the process for the gas.
b) Determine the change in entropy for the gas, using the thermodynamic definition of entropy.
An alternative expression for the entropy is the so-called Sackur-Tetrode equation, that is based
on quantum mechanics:
kT 5 h
S = N k ln 3
e 2 with Λ = √ ,
PΛ 2πmkT
which describes the entropy of a perfect atomic gas. In this expression m is the mass of the atoms,
h is Planck’s constant, e = exp(1) and the other symbols have their usual meaning.
c) Show that the Sackur-Tetrode equation results in the same change in entropy for the gas.
d) Repeat subproblem b) for a compression of the gas by cooling it again from 177◦ C to 127◦ C,
but now irreversibly.
Problem 3
The osmotic pressure for an enzyme in an aqueous solution is determined by measuring the rise of
the solution h in a column (h is the pressure in terms of the rise of a solution column with density
ρ = 1.004 g cm−3 ), at a temperature of 20 ◦ C.
The rise is determined to be h = 5.746 cm.
The molar mass of pure water is M (H2 O) = 18.02 g/mol with a density of ρH2 O = 0.9982 gcm−3
at 20 ◦ C.
a) Calculate the concentration (in mol/L) of the enzyme, assuming an ideal solution.
b) Show that the osmotic pressure of the real (non-ideal) solution is given by
RT
Π=− ∗ ln aA .
Vm,A
For a molality of the enzyme of 1.000 · 10−3 mol/kg a more accurate measurement of the activity
of the water in the solution turns out to be equal to 0.9950.
c) Calculate the osmotic pressure of the solution based on this water activity.
We add another enzyme to the solution with a molality of 0.500 mol/kg. The activity of the water
turns out to be unchanged.
d) Calculate the osmotic pressure of the solution for this new situation.
, 3
Problem 4
Consider a NiCd-battery at T = 298 K. This is a battery which is rechargeable as a result of a
cleverly chosen cathode and anode design, for which the reaction products at discharge are well
collected at the electrodes.
The redox reaction between Cd and Cd(OH)2 (s) has a standard reduction potential of -0.81 V while
the reaction between NiO(OH)(s) and Ni(OH)2 (s), with H2 O as intermediate, has as standard
reduction potential of 0.49 V.
The two electrodes are separated by an electrolyte consisting of an aqueous KOH solution.
The positive electrode contains 2.50 g of Ni (M (Ni) = 58.69 g/mol).
The internal resistance of the cell, Ri , is unknown.
The initial state is a fully charged cell.
In the table below some thermodynamic data at T = 298 K are collected.
a) Give the reactions at both electrodes as well as the total cell reaction, all for discharging
conditions.
We load the fully charged cell at isothermal conditions with an external resistor of RL = 5 Ω.
The discharge process of the cell can in good approximation be described by a constant cell voltage
Ecell , dropping suddenly to zero at the end of the process, when the cell is fully discharged.
With the load resistor, RL , the time needed to discharge the cell fully is 5.3 hour.
b) Determine ∆f G (Cd(OH)2 ) at T = 298 K.
c) Calculate the total electrical work the cell has delivered to RL , once equilibrium is reached.
d) Calculate the efficiency of the discharge process.
OH− (aq) H2 O Cd(s)
∆f G (kJ/mol) -157 -237
∆f H (kJ/mol) -230 -286
Sm (J/molK) -10.8 69.9 51.8
List of constants
Elementary charge e 1.602 · 10−19 C
Faraday’s constant F 9.648 · 104 Cmol−1
Boltzmann’s constant k 1.381 · 10−23 JK−1
Planck’s constant h 6.626 · 10−34 Js
Bohr Magneton µB 9.274 · 10−24 JT−1
Atomic mass constant mu 1.661 · 10−27 kg
Amadeo Avogadro di Quaregna e Ceretto’s constant NA 6.022 · 1023 mol−1
Gas constant R 8.314 JK−1 mol−1
Free fall acceleration g 9.807 ms−2
Unit of energy 1 cal = 4.184 J
Standard pressure P 1 bar = 105 Nm−2 = 0.9869 atm = 750 Torr
Exam Thermodynamics
2 June 2016
Please, hand in your answers to problems 1, 2, 3 and 4 on separate sheets.
Put your name and student number on each sheet.
The examination time is 18:00 until 21:00.
There are 4 problems, with each 4 subproblems, a list of constants and a formulae sheet,
4 pages in total.
All 16 subproblems have equal weight for the final grade.
Problem 1
a) Give a definition and short description of the thermodynamic background for the following
concepts
– partial pressure
– reaction quotient
– ideal solution
– Boltzmann distribution
– work
b) Indicate for each of the following quantities whether we are dealing with a state function (all
symbols have their usual meaning); answer using only yes or no:
– V P,
– P V 2,
– PV − Q + TS − W,
– A,
– Qrev .
c) Give the meaning of all symbols in the following formula as well as a description of its use
in thermodynamic problems.
H − kT ln W
d) A system of molecules has a molecular vibrational energy spectrum given by i = i · , where
= 6 · 10−21 J, and i is a positive integer (i = 0, 1, 2, 3, 4, 5, .....).
At T = 300 K and P = P the probability for a molecule to occupy the vibrational state
labeled i = 7 equals 3.01 · 10−5 .
Calculate the probability for a molecule to occupy the vibrational state with energy 2 at
T = 300 K and P = P .
,2
Problem 2
In this problem we study the entropy change after an isochoric heating process for a perfect atomic
gas. We start the process with 2 mol of the gas in a closed cylinder with a volume V = 2 L, and
we heat the gas from 127◦ C to 177◦ C reversibly.
a) Determine the pressure at the start of the process for the gas.
b) Determine the change in entropy for the gas, using the thermodynamic definition of entropy.
An alternative expression for the entropy is the so-called Sackur-Tetrode equation, that is based
on quantum mechanics:
kT 5 h
S = N k ln 3
e 2 with Λ = √ ,
PΛ 2πmkT
which describes the entropy of a perfect atomic gas. In this expression m is the mass of the atoms,
h is Planck’s constant, e = exp(1) and the other symbols have their usual meaning.
c) Show that the Sackur-Tetrode equation results in the same change in entropy for the gas.
d) Repeat subproblem b) for a compression of the gas by cooling it again from 177◦ C to 127◦ C,
but now irreversibly.
Problem 3
The osmotic pressure for an enzyme in an aqueous solution is determined by measuring the rise of
the solution h in a column (h is the pressure in terms of the rise of a solution column with density
ρ = 1.004 g cm−3 ), at a temperature of 20 ◦ C.
The rise is determined to be h = 5.746 cm.
The molar mass of pure water is M (H2 O) = 18.02 g/mol with a density of ρH2 O = 0.9982 gcm−3
at 20 ◦ C.
a) Calculate the concentration (in mol/L) of the enzyme, assuming an ideal solution.
b) Show that the osmotic pressure of the real (non-ideal) solution is given by
RT
Π=− ∗ ln aA .
Vm,A
For a molality of the enzyme of 1.000 · 10−3 mol/kg a more accurate measurement of the activity
of the water in the solution turns out to be equal to 0.9950.
c) Calculate the osmotic pressure of the solution based on this water activity.
We add another enzyme to the solution with a molality of 0.500 mol/kg. The activity of the water
turns out to be unchanged.
d) Calculate the osmotic pressure of the solution for this new situation.
, 3
Problem 4
Consider a NiCd-battery at T = 298 K. This is a battery which is rechargeable as a result of a
cleverly chosen cathode and anode design, for which the reaction products at discharge are well
collected at the electrodes.
The redox reaction between Cd and Cd(OH)2 (s) has a standard reduction potential of -0.81 V while
the reaction between NiO(OH)(s) and Ni(OH)2 (s), with H2 O as intermediate, has as standard
reduction potential of 0.49 V.
The two electrodes are separated by an electrolyte consisting of an aqueous KOH solution.
The positive electrode contains 2.50 g of Ni (M (Ni) = 58.69 g/mol).
The internal resistance of the cell, Ri , is unknown.
The initial state is a fully charged cell.
In the table below some thermodynamic data at T = 298 K are collected.
a) Give the reactions at both electrodes as well as the total cell reaction, all for discharging
conditions.
We load the fully charged cell at isothermal conditions with an external resistor of RL = 5 Ω.
The discharge process of the cell can in good approximation be described by a constant cell voltage
Ecell , dropping suddenly to zero at the end of the process, when the cell is fully discharged.
With the load resistor, RL , the time needed to discharge the cell fully is 5.3 hour.
b) Determine ∆f G (Cd(OH)2 ) at T = 298 K.
c) Calculate the total electrical work the cell has delivered to RL , once equilibrium is reached.
d) Calculate the efficiency of the discharge process.
OH− (aq) H2 O Cd(s)
∆f G (kJ/mol) -157 -237
∆f H (kJ/mol) -230 -286
Sm (J/molK) -10.8 69.9 51.8
List of constants
Elementary charge e 1.602 · 10−19 C
Faraday’s constant F 9.648 · 104 Cmol−1
Boltzmann’s constant k 1.381 · 10−23 JK−1
Planck’s constant h 6.626 · 10−34 Js
Bohr Magneton µB 9.274 · 10−24 JT−1
Atomic mass constant mu 1.661 · 10−27 kg
Amadeo Avogadro di Quaregna e Ceretto’s constant NA 6.022 · 1023 mol−1
Gas constant R 8.314 JK−1 mol−1
Free fall acceleration g 9.807 ms−2
Unit of energy 1 cal = 4.184 J
Standard pressure P 1 bar = 105 Nm−2 = 0.9869 atm = 750 Torr
