7.4 Heat Capacity of Nanoparticles 141
7.4
Heat Capacity of Nanoparticles
The heat capacity C of a solid object is defined as the absorbed heat ΔE that is
necessary to increase the temperature by ΔT.
C
E
T
=
∆
∆
.
(7.15)
The numbers given in tables are either for 1 g (specific heat capacity) or 1 mol,
(molar heat capacity). Analyzing the behavior of nanoparticles, often the heat
capacity of one particle is taken as a reference. For gases, it is important to take
care whether the heat capacity refers to constant pressure or constant volume; as,
in case of solids, the thermal expansion are comparatively small, this difference
is not significant.
The heat capacity of crystallized solids is well understood. It is the sum of the
energy of lattice vibrations. As, in a crystallized solid, the atoms are in well-defined
regular distances, only a limited number of vibrations, the phonons, is possible.
Certainly, the longer the wavelength, the lower is the energy of the phonons, the
more phonons are active, the higher is the heat capacity. Therefore, as nanoparticles are small, only a smaller number of phonons, as compared to bulk materials,
is possible. Hence, one expects a reduced heat capacity. However, there is a second
effect influencing the heat capacity: In a disordered system like a liquid, for the
atoms there are more degrees of freedom to vibrate, as they are no longer fixed in
a lattice. Therefore, one expects a higher heat capacity. The same, however, to a
reduced extent is observed for amorphous material and, therefore, one may expect
this for grain boundaries, too. As was shown in Chapter 3, in the case of small
nanoparticles, a significant amount of the volume is on the grain boundaries, one
has to expect a significant effect, superimposing or even exceeding the effect of
smallness on the phonon-influenced part of the heat capacity.
Box 7.5 Heat Capacity of a Small Crystal
To elucidate the statement that the heat capacity caused by lattice vibrations
of a small particle should be smaller than that of bulk material, the dependency
of the possible number of phonons on the crystal size is estimated. To clarify
the idea, a linear model of a crystal will be used, for further simplification, it
is assumed that the ends of this crystal are fixed and the number of atoms is
odd. Figure 7.19 displays this model crystal that consists of n atoms, each one
at a distance of a, the lattice parameter. The size of this crystal l is given by
l = (n − 1)a.
As stated in the boundary conditions, vibrations of such a chain have nodes
at its ends. Furthermore, only those vibrations are possible that have nodes at
7.4
Heat Capacity of Nanoparticles
The heat capacity C of a solid object is defined as the absorbed heat ΔE that is
necessary to increase the temperature by ΔT.
C
E
T
=
∆
∆
.
(7.15)
The numbers given in tables are either for 1 g (specific heat capacity) or 1 mol,
(molar heat capacity). Analyzing the behavior of nanoparticles, often the heat
capacity of one particle is taken as a reference. For gases, it is important to take
care whether the heat capacity refers to constant pressure or constant volume; as,
in case of solids, the thermal expansion are comparatively small, this difference
is not significant.
The heat capacity of crystallized solids is well understood. It is the sum of the
energy of lattice vibrations. As, in a crystallized solid, the atoms are in well-defined
regular distances, only a limited number of vibrations, the phonons, is possible.
Certainly, the longer the wavelength, the lower is the energy of the phonons, the
more phonons are active, the higher is the heat capacity. Therefore, as nanoparticles are small, only a smaller number of phonons, as compared to bulk materials,
is possible. Hence, one expects a reduced heat capacity. However, there is a second
effect influencing the heat capacity: In a disordered system like a liquid, for the
atoms there are more degrees of freedom to vibrate, as they are no longer fixed in
a lattice. Therefore, one expects a higher heat capacity. The same, however, to a
reduced extent is observed for amorphous material and, therefore, one may expect
this for grain boundaries, too. As was shown in Chapter 3, in the case of small
nanoparticles, a significant amount of the volume is on the grain boundaries, one
has to expect a significant effect, superimposing or even exceeding the effect of
smallness on the phonon-influenced part of the heat capacity.
Box 7.5 Heat Capacity of a Small Crystal
To elucidate the statement that the heat capacity caused by lattice vibrations
of a small particle should be smaller than that of bulk material, the dependency
of the possible number of phonons on the crystal size is estimated. To clarify
the idea, a linear model of a crystal will be used, for further simplification, it
is assumed that the ends of this crystal are fixed and the number of atoms is
odd. Figure 7.19 displays this model crystal that consists of n atoms, each one
at a distance of a, the lattice parameter. The size of this crystal l is given by
l = (n − 1)a.
As stated in the boundary conditions, vibrations of such a chain have nodes
at its ends. Furthermore, only those vibrations are possible that have nodes at