Factors
that affect specific heat capacity
For any given substance, the heat capacity of a
body is directly proportional to the amount of substance it contains (measured
in terms of mass or moles or volume). Doubling the amount of substance in a
body doubles its heat capacity, etc.
However, when this effect has been corrected for,
by dividing the heat capacity by the quantity of substance in a body, the
resulting specific heat capacity is a function of the
structure of the substance itself. In particular, it depends on the number of degrees of freedom that
are available to the particles in the substance, each of which type of freedom
allows substance particles to store energy. The translational kinetic
energy of substance particles is only one of the many possible degrees of
freedom which manifests as temperature change, and thus the larger the
number of degrees of freedom available to the particles of a substance other
than translational kinetic energy, the larger will be the specific heat
capacity for the substance. For example, rotational kinetic energy of gas
molecules stores heat energy in a way that increases heat capacity, since this
energy does not contribute to temperature.
In addition, quantum effects require that whenever
energy be stored in any mechanism associated with a bound system which confers
a degree of freedom, it must be stored in certain minimal-sized deposits
(quanta) of energy, or else not stored at all. Such effects limit the full
ability of some degrees of freedom to store energy when their lowest energy
storage quantum amount is not easily supplied at the average energy of
particles at a given temperature. In general, for this reason, specific heat
capacities tend to fall at lower temperatures where the average thermal energy
available to each particle degree of freedom is smaller, and thermal energy
storage begins to be limited by these quantum effects. Due to this process, as
temperature falls toward absolute zero, so also does heat capacityDegrees of freedom
Molecules are quite different from the monatomic
gases like helium
and argon.
With monatomic gases, thermal energy comprises only translational motions.
Translational motions are ordinary, whole-body movements in 3D space whereby particles move about and
exchange energy in collisions—like rubber balls in a vigorously shaken
container (see animation here [16]).
These simple movements in the three dimensions of space mean individual atoms
have three translational degrees of freedom. A
degree of freedom is any form of energy in which heat transferred into an
object can be stored. This can be in translational kinetic energy, rotational kinetic energy, or other forms
such as potential energy in vibrational
modes. Only three translational degrees of freedom (corresponding to
the three independent directions in space) are available for any individual
atom, whether it is free, as a monatomic molecule, or bound into a polyatomic
molecule.
As to rotation about an atom's axis (again,
whether the atom is bound or free), its energy of rotation is proportional to
the moment of inertia for the atom, which is
extremely small compared to moments of inertia of collections of atoms. This is
because almost all of the mass of a single atom is concentrated in its nucleus,
which has a radius too small to give a significant moment
of inertia. In contrast, the spacing of quantum energy levels for a
rotating object is inversely proportional to its moment of inertia, and so this
spacing becomes very large for objects with very small moments of inertia. For
these reasons, the contribution from rotation of atoms on their axes is
essentially zero in monatomic gases, because the energy spacing of the
associated quantum levels is too large for significant thermal energy to be
stored in rotation of systems with such small moments of inertia. For similar
reasons, axial rotation around bonds joining atoms in diatomic gases (or along
the linear axis in a linear molecule of any length) can also be neglected as a
possible "degree of freedom" as well, since such rotation is similar
to rotation of monatomic atoms, and so occurs about an axis with a moment of
inertia too small to be able to store significant heat energy.
In polyatomic molecules, other rotational modes
may become active, due to the much higher moments of inertia about certain axes
which do not coincide with the linear axis of a linear molecule. These modes
take the place of some translational degrees of freedom for individual atoms,
since the atoms are moving in 3-D space, as the molecule rotates. The narrowing
of quantum mechanically determined energy spacing between rotational states
results from situations where atoms are rotating around an axis that does not
connect them, and thus form an assembly that has a large moment of inertia.
This small difference between energy states allows the kinetic energy of this
type of rotational motion to store heat energy at ambient temperatures.
Furthermore internal vibrational degrees of freedom also may become active
(these are also a type of translation, as seen from the view of each atom). In
summary, molecules are complex objects with a population of atoms that may move
about within the molecule in a number of different ways (see animation at
right), and each of these ways of moving is capable of storing energy if the
temperature is sufficient.
The heat capacity of molecular substances (on a
"per-atom" or atom-molar, basis) does not exceed the heat capacity of
monatomic gases, unless vibrational modes are brought into play. The reason for
this is that vibrational modes allow energy to be stored as potential energy in
intra-atomic bonds in a molecule, which are not available to atoms in monatomic
gases. Up to about twice as much energy (on a per-atom basis) per unit of
temperature increase can be stored in a solid as in a monatomic gas, by this
mechanism of storing energy in the potentials of interatomic bonds. This gives
many solids about twice the atom-molar heat capacity at room temperature of
monatomic gases.
However, quantum effects heavily affect the actual
ratio at lower temperatures (i.e., much lower than the melting temperature of
the solid), especially in solids with light and tightly bound atoms (e.g.,
beryllium metal or diamond). Polyatomic gases store intermediate amounts of
energy, giving them a "per-atom" heat capacity that is between that of
monatomic gases (3⁄2 R per
mole of atoms, where R is the ideal gas constant), and the maximum of fully
excited warmer solids (3 R per mole of atoms). For gases, heat capacity
never falls below the minimum of 3⁄2
R per mole (of molecules), since the kinetic energy of gas molecules is
always available to store at least this much thermal energy. However, at
cryogenic temperatures in solids, heat capacity falls toward zero, as
temperature approaches absolute zero.Table of specific heat capacities
Note that the especially high molar values, as for paraffin,
gasoline, water and ammonia, result from calculating specific heats in terms of
moles of molecules. If specific heat is expressed per mole of atoms
for these substances, none of the constant-volume values exceed, to any large
extent, the theoretical Dulong-Petit limit of 25 J·mol−1·K−1
= 3 R per mole of atoms (see the last column of this table). Paraffin, for
example, has very large molecules and thus a high heat capacity per mole, but
as a substance it does not have remarkable heat capacity in terms of volume,
mass, or atom-mol (which is just 1.41 R per mole of atoms, or less than half of
most solids, in terms of heat capacity per atom).
In the last column, major departures of solids at
standard temperatures from the Dulong-Petit
law value of 3 R, are usually due to low atomic weight plus high
bond strength (as in diamond) causing some vibration modes to have too much
energy to be available to store thermal energy at the measured temperature. For
gases, departure from 3 R per mole of atoms in this table is generally due to
two factors: (1) failure of the
higher quantum-energy-spaced vibration modes in gas molecules to be excited at
room temperature, and (2) loss
of potential energy degree of freedom for small gas molecules, simply because
most of their atoms are not bonded maximally in space to other atoms, as
happens in many solids|
Notable minima and
maxima are shown in maroon
|
||||||
|
Table of specific
heat capacities at 25 °C (298 K) unless otherwise noted.[citation
needed]
Substance
|
Isobaric
mass heat capacity cP J·g−1·K−1 |
Isobaric
molar heat capacity CP,m J·mol−1·K−1 |
Isochore
molar heat capacity CV,m J·mol−1·K−1 |
|||
|
Air (Sea level, dry,
0 °C (273.15 K)) |
gas
|
1.0035
|
29.07
|
20.7643
|
0.001297
|
~ 1.25 R
|
|
Air (typical
room conditionsA) |
gas
|
1.012
|
29.19
|
20.85
|
0.00121
|
~ 1.25 R
|
|
solid
|
0.897
|
24.2
|
2.422
|
2.91 R
|
||
|
liquid
|
4.700
|
80.08
|
3.263
|
3.21 R
|
||
|
mixed
|
3.5
|
3.7*
|
||||
|
solid
|
0.207
|
25.2
|
1.386
|
3.03 R
|
||
|
gas
|
0.5203
|
20.7862
|
12.4717
|
1.50 R
|
||
|
solid
|
0.328
|
24.6
|
1.878
|
2.96 R
|
||
|
solid
|
1.82
|
16.4
|
3.367
|
1.97 R
|
||
|
solid
|
0.123
|
25.7
|
1.20
|
3.09 R
|
||
|
solid
|
0.231
|
26.02
|
3.13 R
|
|||
|
gas
|
0.839*
|
36.94
|
28.46
|
1.14 R
|
||
|
solid
|
0.449
|
23.35
|
2.81 R
|
|||
|
solid
|
0.385
|
24.47
|
3.45
|
2.94 R
|
||
|
solid
|
0.5091
|
6.115
|
1.782
|
0.74 R
|
||
|
liquid
|
2.44
|
112
|
1.925
|
1.50 R
|
||
|
Gasoline (octane)
|
liquid
|
2.22
|
228
|
1.64
|
1.05 R
|
|
|
solid
|
0.84
|
|||||
|
solid
|
0.129
|
25.42
|
2.492
|
3.05 R
|
||
|
solid
|
0.790
|
2.17
|
||||
|
solid
|
0.710
|
8.53
|
1.534
|
1.03 R
|
||
|
gas
|
5.1932
|
20.7862
|
12.4717
|
1.50 R
|
||
|
gas
|
14.30
|
28.82
|
1.23 R
|
|||
|
gas
|
1.015*
|
34.60
|
1.05 R
|
|||
|
solid
|
0.450
|
25.09[29]
|
3.537
|
3.02 R
|
||
|
solid
|
0.129
|
26.4
|
1.44
|
3.18 R
|
||
|
solid
|
3.58
|
24.8
|
1.912
|
2.98 R
|
||
|
liquid
|
4.379
|
30.33
|
2.242
|
3.65 R
|
||
|
solid
|
1.02
|
24.9
|
1.773
|
2.99 R
|
||
|
liquid
|
0.1395
|
27.98
|
1.888
|
3.36 R
|
||
|
Methane at 2 °C
|
gas
|
2.191
|
35.69
|
0.66 R? 4.23R
|
||
|
liquid
|
2.14
|
68.62
|
1.38 R
|
|||
|
Molten salt (142-540 °C)[32]
|
liquid
|
1.56
|
2.62
|
|||
|
gas
|
1.040
|
29.12
|
20.8
|
1.25 R
|
||
|
gas
|
1.0301
|
20.7862
|
12.4717
|
1.50 R
|
||
|
gas
|
0.918
|
29.38
|
21.0
|
1.26 R
|
||
|
Paraffin wax
C25H52 |
solid
|
2.5 (ave)
|
900
|
2.325
|
1.41 R
|
|
|
solid
|
2.3027
|
|||||
|
Silica (fused)
|
solid
|
0.703
|
42.2
|
1.547
|
1.69 R
|
|
|
solid
|
0.233
|
24.9
|
2.44
|
2.99 R
|
||
|
solid
|
1.230
|
28.23
|
3.39 R
|
|||
|
solid
|
0.466
|
|||||
|
solid
|
0.227
|
27.112
|
3.26 R
|
|||
|
solid
|
0.523
|
26.060
|
3.13 R
|
|||
|
solid
|
0.134
|
24.8
|
2.58
|
2.98 R
|
||
|
solid
|
0.116
|
27.7
|
2.216
|
3.33 R
|
||
|
Water at 100 °C (steam)
|
gas
|
2.080
|
37.47
|
28.03
|
1.12 R
|
|
|
Water at 25 °C
|
liquid
|
4.1813
|
75.327
|
74.53
|
4.1796
|
3.02 R
|
|
Water at 100 °C
|
liquid
|
4.1813
|
75.327
|
74.53
|
4.2160
|
3.02 R
|
|
solid
|
2.11
|
38.09
|
1.938
|
1.53 R
|
||
|
solid
|
0.387
|
25.2
|
2.76
|
3.03 R
|
||
|
Substance
|
Isobaric
mass heat capacity cP J·g−1·K−1 |
Isobaric
molar heat capacity CP,m J·mol−1·K−1 |
Isochore
molar heat capacity CV,m J·mol−1·K−1 |
|
||
http://en.wikipedia.org/wiki/Heat_capacity
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