Download 7. Heat capacity

7. Heat capacity
Basel, 2008
7. Heat capacity
1.
Introduction
2.
Heat capacity at constant volume
3.
Internal energy
4.
Classic approach – gases (Internal energy/heat capacity)
5.
Quantum mechanics approach – gases (Internal energy/heat
capacity)
References:
Supplementary material:
German version for this
chapter (Prof. Huber lecture
from 2007, Kapitel 7).
Web: Maple-Sheet „Cv-Rotat.“
•P. Atkins, J. de Paula, “Atkins‘ Physical
Chemistry”, Oxford Univ. Press, Oxford, 8th
ed., 2006, Chapter 2 and 20.
• I. Tinoco, K. Sauer, J.C. Wang, J.D. Puglisi
“Physical
Chemistry,
Principles
and
applications in biological sciences”, PrenticeHall, New Jersey, 4th ed., 2002, Chapter 2.
7.1 Introduction
Heat capacity of the local surroundings of a system is essential in
thermodynamics, because the heat absorbed or evolved by the system can
be monitored as a temperature change.
When heat is transferred to a system
(proportional to the heat supplied):
>
change in temperature,
T – temperature
dT = Coefficient × dq
(6.1)
q - heat
Heat capacity, C is a measure of the heat energy required to increase the
temperature of a substance, by a certain temperature interval (specific heat),
which depends on the size, composition and state of the system.
dq = C × dT
(6.2)
Heat capacity
Molar heat capacity:
Cm = C/n
(6.3)
n – number of moles
M – mass in kg
Specific heat capacity:
c = C/M
(6.4)
When C is large > a given amount of heating results in only a small
temperature rise (the system has a large capacity for heat)
The heat capacity depends on conditions:
- system constrained to have constant volume > Cv (heat capacity at
constant volume, or isochoric heat capacity)
- system subject to constant pressure > Cp (heat capacity at constant
pressure, or isobaric heat capacity)
7.2 Heat capacity at constant volume
Heat capacity at constant volume represents the heat supplied to the
system under specific conditions (constant volume and no other work).
It can be related to the increase in internal energy that accompanies the
heating:
δ- partial derivatives with respect to the
(6.5)
δ
U
⎞
⎛
CV = ⎜
⎟
⎝ δT ⎠V
changing variable (when one or more variables
are held constant during the change of another
variable)
U – internal energy of the system
T - temperature
Molar heat capacity
constant volume:
CV ,m
⎛ δU m ⎞
=⎜
⎟
⎝ δT ⎠V
at
Specific heat capacity at
constant volume:
(6.6)
Um – molar internal energy
[CV,m ] = J/(mol K)
1
cV =
M
⎛ δU ⎞
⎟
⎜
⎝ δT ⎠V
(6.7)
[cV ] = J/(kg K)
7.3 Internal energy of the system
Internal energy, U: total kinetic and potential energy of the particles in the
system (summ of all contributions from all atoms, ions, and molecules in the
system).
U = Uo + Utrans + Urot + Uvib + Uel
translation
rotation
(6.8)
U0 – internal energy at T = 0
[U] = J = 1 kg m2s-2
electronic
vibration
1 cal = 4.184 J
1 eV = 0.16 10-18J
Characteristics of the internal energy:
• U = f(T, p)
• U is an extensive property of the system
Internal energy – function of state
U is a function of state (depends only on the present state of the system
and is independent on the path by which the state was reached).
A property which depends on two variables is
function of state if it depends only on the current
state of the system.
∆U is independent on the
path between these states.
ab
le
2
P0
Va
ri
A change in this property is independent on the
path between the initial state (P0), and the final
state (Pf).
(6.9)
∆U = Uf - Ui
Pf
Variable 1
Convention: ∆U > 0 – a flow of energy into the system
∆U < 0 – a flow of energy out of the system
Internal energy of the system
Total internal energy of a system – too complicate to measure it directly >
∆U can be measured (it is possible to monitor the energy supplied or lost by
the system, as heat or work:
∆U = w + q
(6.10)
w – energy transfered to the sytem by
dooing work
q – energy transfered to the system, by
heating
Heat capacity is a summ of various terms according to the type of energy
which is contained in the internal energy of the system.
C = Ctrans + Crot + Cvib + Cel
(6.11)
7.4 Classic approach - gases
Degree of freedom of a molecule (3N):
- Translation (3)
- Rotation (linear molecules: nrot = 2, non-linear molecules nrot = 3)
- Vibration (linear molecules nvib = 3N-5, non-linear molecules nvib = 3N-6)
Internal energy of a molecule, due to translation (kinetic energy):
U tr =
(
)
1
3
m v 2x + v 2y + v 2z = kT
2
2
(6.12)
Equipartition theorem: for a collection of particles in thermal equilibrium (T),
the average value of each quadratic contribution to the energy is the same = ½
kT (or ½ RT if we consider the molar contribution).
Total energy of the gas of N particles (considering only the translation):
U = U0 +
3
NkT
2
(6.13)
7.4.1 Total internal energy of a gas
Total molar energy of a gas (considering only the translation movement):
U m ,tr = U m (0) +
3
RT
2
(6.13)
Um
N = nNA R = Nak
k – Boltzmann constant k = 1.38 10-23 JK-1
T
Um, tr – linear in temperature
Consider the rotation moverment of molecules: for each degree of freedom of
rotation a contribution of 1/2kT (or ½ RT for molar internal energy)
Total molar internal energy of the system, considering the translation and
rotation:
U m ,tr
3
1
= U m (0) + RT + nrot RT
2
2
(6.14)
7.4.2 Molar heat capacity at constant
volume
Total internal molar energy of the system, considering in addition the
vibration movement:
U m ,tr
3
1
= U m (0) + RT + nrot RT + nvib RT
2
2
(6.15)
In vibration degree of
freedom
should
be
considered a potential
energy term, of 1/2RT
Molar heat capacity at constant volume:
Cv , m
⎛ δU m ⎞ ⎛ 3 1
⎞
=⎜
⎟ = ⎜ + nrot + nvib ⎟ R
⎠
⎝ δT ⎠ ⎝ 2 2
Cv, m
(6.16)
Do not depend on
temperature
T
Molar heat capacity at constant volume
Example: each HCl molecule of
gas changes its movement due to
the collisions with other molecules
in the system (coloured points)
Rotation
Translation
H
Cv , m
7
⎛3 1
⎞
= ⎜ + nrot + nvib ⎟ R = R
2
⎝2 2
⎠
Cl
Vibration
Mass center
nrot = 2, nvib=1
Cv,m = 29.10 J mol-1 K-1 > 20.6 J mol-1 K-1 (experimental value, at 300K)
Quantum mechanics approach
7.5 Quantum mechanics approach
A gas system formed by diatomic molecules at temperature T has a
vibrational energy which is the summ of all vibrational contributions of the
molecules.
1. Each molecule has a vibrational energy (harmonic oscillator approx.):
1⎞
⎛
Evib (v ) = ⎜ v + ⎟hν vib
2⎠
⎝
(6.17)
2. The v-th energy level is occupied by n(v) molecules, those number
corresponds to the Boltzmann distribution:
n(v ) = n0 e
− Evib ( v )
(6.18)
kT
7.5.1 Vibrational energy of a gas
Total vibrational energy of the gas = summ of (all energies multiplied
with the Boltzmann occupancy of every energy level)
ET = ∑ E (v i ) =
i
5
+ hν ⋅ n0 e
2
5
− hν
2
1
hν ⋅ n0 e
2
kT
1
− hν
2
kT
3
+ hν ⋅ n0 e
2
⎛ 1⎞
+ K + ⎜ i + ⎟hν ⋅ n0 e
⎝ 2⎠
3
− hν
2
⎛ 1⎞
− ⎜ i + ⎟ hν
⎝ 2⎠
kT
kT
+
(6.19)
+K
Total number of molecules in the system = summ of molecules occupying
each vibrational energy level:
N T = ∑ n(v i ) = n0 e
1
− hν
2
kT
+ n0 e
3
− hν
2
kT
i
+ n0 e
5
− hν
2
kT
+ K + n0 e
⎛ 1⎞
− ⎜ i + ⎟ hν
⎝ 2⎠
kT
+K
+
(6.20)
Mean vibrational energy of a molecule
The mean vibrational energy of a gas molecule = total vibrational energy of
the gas/total number of gas molecules (for complete calculus see: K7-4,
Vorlesungsskript, 2007, H. Huber).
∑ E (v ) ⎡ 1 e ν
=
= hν ⎢ +
ν
(
)
n
v
2
⎢⎣
∑
1− e
−h
i
Evib ,molecule
kT
i
−h
i
i
⎤
⎥
kT ⎥
⎦
(6.21)
Molar internal energy due to the vibration is:
U vib (T ) = N A h ν ⋅
[ + (e
1
2
h ν / kT
− 1)
−1
]= R ⋅ Θ ( + (e
vib
1
2
Θ vib /T
− 1)
Θvib = hν/k
−1
)
(6.22)
7.5.2 Heat capacity term - vibration
The term of heat capacity corresponding to a mole of gas, due to the
vibration movement is:
C vib (T ) = R ⋅ (hν / kT ) e
2
hν /kT
C vib (T ) = R ⋅ (Θ vib / T ) e
2
- Characteristic temperature:
- Low temperatures (T = 0 K)
- High temperatures (T >> Θ)
(e
hν /kT
−Θ vib /T
− 1)
−2
(1 − e
T=Θ
(6.23)
−Θ vib /T
)
2
(6.24)
Cvib(T) = R(e -1)2/e3
Uvib = ½ NA h ν
Uvib
RT
Cvib = 0
Cvib
R
7.5.3 Mean rotational energy of a
molecule
The mean rotational energy of a gas molecule = total rotational energy of
the gas/total number of gas molecules:
∑N E
=
∑N
i
i
Erot ,molecule
i
i
i
− hν
⎡1
e kT ⎤
⎥
= hν ⎢ +
− hν
⎢⎣ 2 1 − e kT ⎥⎦
(6.25)
Internal energy due to the rotation movement:
Urot = NA B/q Σ (2J+1)J(J+1) eBJ(J+1)/(kT)
q= Σ(2J+1)e-BJ(J+1)/kT
(6.27)
(6.26)
7.5.4 Heat capacity term - rotation
Molar heat capacity term at constant
volume, due to the rotation of the
molecules in the gas:
Cv ,m
⎛ δU
= ⎜⎜ m ,rot
⎝ δT
⎞
⎟⎟
⎠
Cv, rot
(6.28)
T
To understand and learn
- Wie ist die Molwärme bei konstantem Volumen definiert?
- Was geben die Molwärmen an (in Worten)?
- Was besagt der klassische Gleichverteilungssatz?
- Wie gross ist klassisch die innere Energie des idealen Gases (ausgedrückt durch die
Freiheitsgrade)?
- Wie gross ist klassisch Cm,V des idealen Gases (ausgedrückt durch die Freiheitsgrade)?
- -Wie gross ist Cm,V von Helium bei 300 K?
- Wie gross ist Cm,V von HCl bei 300 K etwa?
- Was bedeutet die Aussage, ein Vibrations-Freiheitsgrad sei aktiv bzw. inaktiv?
- Welche Freiheitsgrade sind bei Zimmertemperatur typisch aktiv, inaktiv bzw. teilweise
aktiv?
- Wie lässt sich damit aussagen, wieweit ein Freiheitsgrad aktiv oder inaktiv ist?
- Zeichnen Sie den Verlauf von Cm,V als Funktion der Temperatur für ein zweiatomiges Gas!
- Skizzieren Sie das Vorgehen zur Berechnung der vibratorischen inneren Energie bei
Quantelung!
- Zeichnen Sie U, bzw. CV für eine Vibration klassisch bzw. bei Quantelung als Funktion von
T!
- Skizzieren Sie das Vorgehen zur Berechnung der rotatorischen inneren Energie bei
Quantelung!