Heat capacities

2.4.1 Heat capacities of gases

We have seen in the last section that, if we try to determine changes in energy or enthalpy from experimentally detected changes in temperature, the respective heat capacities are needed. In experimental practice, these are typically determined by calibration. For pure systems, however, it is possible to predict the heat capacities based on physical models of the microscopic effect of increasing heat on the single particle motion.

For ideal gases, we already mentioned that the difference between isochor and isobar heat capacities in case of 1 Mol particles is simply the gas constant R, which is easily derived as following:

ܳ݌ൌ οܪ_݌ ൌ οܷ_݌൅ ݌οܸ ൌ ܿ_ܸ ή οܶ_݌൅ ܴ ή οܶ_݌ ൌ ሺܿ_ܸ൅ ܴሻ ή οܶ_݌ ൌ ܿ_݌ή οܶ_݌ (Eq.2.63) or

ሺܿ_ܸ ൅ ܴሻ ൌ ܿ_݌ (Eq.2.64)

To get an estimate for C_v, we consider that a gas molecule consisting of N atoms is described by 3N Cartesian coordinates x ,y and z, which gives, for the microscopic motion of a molecule, 3N different degrees of freedom. 3 always are translation of the whole molecule in x, y and z direction, 3 more are rotation for non-linear molecules, whereas linear molecules only have 2 rotational degrees of freedom because there is no momentum of inertia if this type of molecule is rotating around its molecular main axis.

The remaining 3N – 6 or 3N – 5 degrees of freedom, respectively, are attributed to molecular vibrations.

Importantly, each degree of freedom of translation or rotation contributes with R/2 to the molar heat capacity , while a vibrational degree of freedom contributes with R. This difference can be explained, in a simplified way, by the different types of energy: translation and rotation are purely kinetic energy, whereas vibration contains both kinetic and potential energy, since it describes the motion of atoms in a force field. As an illustrative example, let us consider the two 3-atomic molecules CO₂ and H₂O.

CO₂: H₂O:

Degrees of freedom: Translation: 3 3

Rotation: 2 3

Vibration: 9 – 5 = 4 9 – 6 = 3

Therefore, the total molar heat capacity of each species is given as:

ܿ_ܸǡܥܱʹ ൌ ͵ ή^ܴ_ʹ൅ ʹ ή^ܴ_ʹ൅ Ͷ ή ܴ ൌ ͸Ǥͷ ή ܴ (Eq.2.65)

ܿ_ܸǡܪʹܱ ൌ ͵ ή^ܴ_ʹ൅ ͵ ή^ܴ_ʹ൅ ͵ ή ܴ ൌ ͸ ή ܴ (Eq.2.66)

The heat capacities differ by nearly 10% just because one rotational degree of freedom is transferred to a vibrational one, if you go from H₂O to CO₂.

Importantly, these heat capacities are limiting values reached at high sample temperature. To address the vibrational degrees of freedom, you need a certain sample temperature already, typically several 100–1000 K. Rotation is more easily addressed already at room temperature, whereas translational motion is for free, i.e. the 3 translational degrees of freedom are excited already at the evaporation temperature irrespective of its value. As a consequence, the heat capacity of the gas increases stepwise with temperature, and the position of the steps or characteristic excitation temperatures depend on specific molecular parameters such as mass of atoms, bond lengths and bond strengths.

7 F 79

Fig. 2.14: T-dependence of the heat capacity of gaseous molecules

2.4.2 Heat capacities of solids

The atoms within a solid cannot move nor rotate but only undergo vibrational motion. We therefore expect a limiting heat capacity of 3R (law of Dulong-Petit), since for a solid consisting of N atoms all 3N degrees of freedom correspond to vibrations, and therefore contribute with R per Mole to the heat capacity.

ܿ_{ܸǡ݉݋݈ܽݎ}ሺܶ ՜ λሻ ൌ ͵ܴ (Eq.2.67)

However, we have learned in the last section that vibrational modes are not accessible at very low temperature. Therefore, experimentally the heat capacity of solids is given by the following expression (Debye):

ܿ_{ܸǡ݉݋݈ܽݎ}ሺܶሻ̱ ቀ_ȣ^ܶ

ቁ^͵ (Eq.2.68)

with ȣ the so-called Debye-temperature that is characteristic for a given solid material.

Basic Physical Chemistry

49

Thermodynamics The thermal accessibility of the vibrations of a solid has theoretically first been addressed by Einstein, who assumed that all atoms have the same oscillatory frequency. According to the Boltzmann factor (see above, section 2.2.2) one then would expect a molar internal energy given as

ܷ ൌ ͵ܰ_ܮ ή ݄ ή ߥ_ܧ ή ቈ ^{‡š’ ቀെ݄ήߥܧ݇ήܶቁ}

ͳെ‡š’ ቀെ^݄ήߥܧ_݇ήܶቁ቉ (Eq.2.69)

And correspondingly the heat capacity is the temperature-derivative, i.e.

ܿ_{ܸǡ݉݋݈ܽݎ} ൌ ቀ^ܷ݀_݀ܶቁ ൌ ͵ܴ ή ቀ^݄ήߥ_݇ήܶ^ܧቁ^ʹή ൥ ^{‡š’ ቀെ݄ήߥܧ݇ήܶቁ}

ቂͳെ‡š’ ቀെ^݄ήߥܧ_݇ήܶቁቃ^ʹ൩ (Eq.2.70)

At very high temperature, this complicated expression yields (Taylor-series-expansion of the exponential!)

ܸܿǡ݉݋݈ܽݎሺܶ ՜ λሻ ൌ ͵ܴ ή ቀ^݄ήߥ_݇ήܶ^ܧቁ^ʹή ൥ ^{ͳെ݄ήߥܧ݇ήܶ}

ቂͳെͳ൅^݄ήߥܧ_݇ήܶቃ^ʹ൩ ൎ ͵ܴ (Eq.2.71)

According to Eq.2.71, the Einstein model fulfills the Dulong-Petit-law valid for every solid consisting of 1 Mole atoms at very high sample temperature. However, at low temperature the molar heat capacity predicted by Einstein is smaller than the one experimentally found.

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The model by Debye describes the experimental result. In contrast to Einstein, he assumes that the vibrations within a solid cover a whole spectrum of different oscillatory frequencies, starting from zero to ߥܧ with the probability (or better: degrees of freedom) of a given frequency as:

ܲሺߥሻ ൌ ͻܰ ή_ߥ^ߥʹ

ܧ͵ (Eq.2.72)

Note that the total number of degrees of freedom, given as the integral of ܲሺߥሻ, equals 3N. This distribution of frequencies is plotted in figure 2.15:

3 Q

PD[ Q

Q

Fig. 2.15: frequency spectrum of the oscillations of a solid

The consequence of this improved model developed by Debye is that oscillations of lower frequencies are already accessible at lower temperatures. Therefore, the heat capacity at lower temperatures is larger than predicted by Einstein, and in agreement with the experimental results. Physically, the existence of lower frequencies is feasible if we consider the coupling of multiple atoms to one oscillator. The frequency of a harmonic oscillator depends on the mass which is moving, and the force which is constraining this movement. If you consider the coupling of atoms to an oscillating aggregate, the mass scales with the total number of atoms whereas the force only scales with the number of surface atoms. Consequently, the oscillatory frequency will decrease with increasing number of coupled atoms, vibration of the whole solid body showing the smallest frequency possible.