2.11: Molar Heat Capacities

Introduction

Heat capacity at constant volume \(C_V\), is defined as

\[ C_V = \left(\dfrac{\partial U}{\partial T} \right)_V, \label{CV_class} \]

and heat capacity at constant pressure \(C_P\), is defined as

\[ C_P = \left(\dfrac{\partial H}{\partial T} \right)_P. \label{CP_class} \]

The statistical definitions of U and H in terms of the canonical partition function Q are:

\[ U = {k_B T^2}\left(\frac{\partial lnQ}{\partial T}\right)_V + U_o\label{UstatQ}\]

and

\[H = k_B T^2\left( \frac{\partial \ln Q}{\partial T} \right)_V + Vk_B T \left( \frac{\partial \ln Q}{\partial V} \right)_T + H_o. \label{HstatQ}\]

Substituting into equations \(\ref{CV_class}\) and \(\ref{CP_class}\) we get statistical expressions for for the heat capacities:

\[ C_V = \left(\dfrac{\partial}{\partial T} \right)_V\left({k_B T^2}\left(\frac{\partial lnQ}{\partial T}\right)_V + U_o\right) = \left(\dfrac{\partial}{\partial T} \right)_V\left({k_B T^2}\left(\frac{\partial lnQ}{\partial T}\right)_V\right),  \label{CV_stat} \]

and

\[ C_P = \left(\dfrac{\partial }{\partial T} \right)_P\left(k_B T^2\left( \frac{\partial \ln Q}{\partial T} \right)_V + Vk_B T \left( \frac{\partial \ln Q}{\partial V} \right)_T + H_o\right) = \left(\dfrac{\partial }{\partial T} \right)_P\left(k_B T^2\left( \frac{\partial \ln Q}{\partial T} \right)_V + Vk_B T \left( \frac{\partial \ln Q}{\partial V} \right)_T\right) . \label{CP_stat} \]

These expressions are general and apply to molecules that have no intermolecular interactions as well as those that are strongly interacting. Calculating Q for interacting particles is challenging (see for example the discussion on the impact of interactions on just the translational contribution to the partition function), so we will limit ourselves to some specific cases where the interactions are unimportant or easier to predict.

We will consider pure samples, so all the particles are identical. If the sample is gas phase this means that:

\[Q(gas) = \frac{q_{trans}^N q_{rot}^N q_{vib}^N q_{elec}^N}{N!},\label{Qgas}\]

where N = the number of molecules in the sample, \(q_{trans}\) is the microcanonical translational partition function, \(q_{rot}\) is the microcanonical rotational partition function, \(q_{vib}\) is the microcanonical vibrational partition function and \(q_{elec}\) is the microcanonical electronic partition function. In the solid phase where the molecules are distinguishable by their location the 1/N! is dropped and translation and rotation are not possible:

\[Q(solid) = {q_{vib}^N q_{elec}^N}.\label{Qsolid}\]

Heat capacity of gases

Equations  \(\ref{CV_stat}\) and  \(\ref{CP_stat}\) depend on the lnQ. So taking the ln of equation \(\ref{Qgas}\) we get:

\[lnQ(gas) = Nlnq_{trans}+ Nlnq_{rot}+Nln q_{vib}+Nlnq_{elec} - lnN!.\label{lnQgas}\]

So,

\[C_V = \left(\dfrac{\partial}{\partial T} \right)_V\left({k_B T^2}\left(\frac{\partial lnQ}{\partial T}\right)_V\right) =\left(\dfrac{\partial}{\partial T} \right)_V\left({k_B T^2}\left(\frac{\partial }{\partial T}\right)_V \left(Nlnq_{trans}+ Nlnq_{rot}+Nln q_{vib}+Nlnq_{elec} - lnN!\right)\right) \\
= \left(\dfrac{\partial}{\partial T} \right)_V\left({k_B T^2} \left(N\left(\frac{\partial lnq_{trans}}{\partial T}\right)_V+ N\left(\dfrac{\partial lnq_{rot}}{\partial T} \right)_V+N\left(\dfrac{\partial ln q_{vib}}{\partial T} \right)_V+N\left(\dfrac{\partial lnq_{elec}}{\partial T} \right)_V - 0\right)\right) \\
=  \left(\dfrac{\partial}{\partial T} \right)_V  N\left[{k_B T^2}\left(\frac{\partial lnq_{trans}}{\partial T}\right)_V+ {k_B T^2}\left(\dfrac{\partial lnq_{rot}}{\partial T} \right)_V+{k_B T^2}\left(\dfrac{\partial ln q_{vib}}{\partial T} \right)_V+{k_B T^2}\left(\dfrac{\partial lnq_{elec}}{\partial T} \right)_V\right].  \label{CV_q}\]

Each term in the square brackets is just the average energy of each particular type of molecular motion (\(\langle E_k\rangle\), where k = trans, rot, vib or elec). In our discussion of equipartition of energy among the different types of molecular motion we developed expressions for these if the molecules do not interact strongly. Inserting the expressions from that discussion converts equation \(\ref{CV_q}\) to:

\[C_V =   \left(\dfrac{\partial}{\partial T} \right)_V  N \left[\frac{3}{2}k_B T+ \frac{D_{rot}}{2} k_B T+D_{vib}^* k_B T \right],  \label{CV_ep1}\]

where \(D_{rot}\) is the number of rotational degrees of freedom (2 for linear and 3 for nonlinear molecules) and \(D_{vib}^*\) is the number of active (low frequency) vibrational degrees of freedom. We have also assumed that the electronic states are far enough apart that \(ln q_{elec}\) is a constant, giving a derivative of zero.  Setting N = N_A (Avogadro's number) and taking the remaining derivative with respect to T gives us an equipartition expression for the molar constant volume heat capacity of a gas:

\[C_{Vm} =   \frac{3}{2}R+ \frac{D_{rot}}{2} R+D_{vib}^* R  = \left( \frac{3}{2}+ \frac{D_{rot}}{2}+D_{vib}^* \right)R,  \label{CVm_ep}\]

where we have used \(N_A k_B = R\), the gas constant. Additionally, we know that for an ideal gas that C_Pm = C_Vm + R giving us an equivalent equipartition expression for the molar constant pressure heat capacity of a gas:

\[C_{Pm} =    \left( \frac{5}{2}+ \frac{D_{rot}}{2}+D_{vib}^* \right)R,  \label{CPm_ep}\]

Thus we predict that:

• A monatomic gas (no rotations and no vibrations) will have a C_Vm = (3/2)R = 12.47 J mol^-1 K^-1 and C_Pm =  (5/2)R = 20.79 J mol^-1 K^-1.
• A linear molecule with no low frequency vibrations will have a C_Vm = (5/2)R = 20.79 J mol^-1 K^-1 and C_Pm =  (7/2)R = 29.10 J mol^-1 K^-1.
• A nonlinear molecule with no low frequency vibrations will have a C_Vm = 3R = 24.94 J mol^-1 K^-1 and C_Pm =  4R = 33.26 J mol^-1 K^-1.
• If some vibrations are active C_Vm will be increased by \(D_{vib}^*R\).

The table below collects some heat capacities for a selection of gases under standard conditions (298 K). You will notice that these simple predictions are very good in many cases

Table \(\PageIndex{1}\): Heat capacities for a selection of gases under standard conditions (298 K). Monatomic gases are in blue, linear molecules in red and nonlinear molecules in purple. Values from the resource table Extended Thermodynamic Properties and the CoolProp software package.

 The experimental values are generally higher than the estimates, except for the case of H₂, which has such a large rotational constant that at room temperature only some of the lowest excited rotational states are accessible. This leads to even the rotational degrees of freedom of H₂ to being less then completely "active". Making a good estimate for H₂ requires using the full summation for q_rot. 

Some other discrepancies to notice:

• Xe which has significant dispersion attractions has a higher heat capacity then predicted for a monatomic gas.
• Other than H₂ linear molecules have heat capacities above the prediction. The heavier the atoms in the molecule, the lower the vibrational frequencies and the more vibrations contribute to the heat capacity.
• For nonlinear molecules, again, the heavier the atoms in the molecule, the lower the vibrational frequencies and the more vibrations contribute to the heat capacity.

Despite these discrepancies, the agreement is very good for such a simple model. Switching to explicit summation to calculate the partitions functions would make the predictions for molecules without significant attractions (non-polar molecules with relatively weak dispersion interactions) quite close.

Temperature dependence of gas phase heat capacities

Our simple model (equation \(\ref{CPm_ep}\)) assumes the rotational degrees of freedom contribute to the heat capacity. However, k_BT has to be much greater than the spacing between the quantum energy levels for this to be true. If this is not satisfied, the heat capacity will be reduced. If the gaseous state could be maintained down to near absolute zero even the translational degrees of freedom would not contribute and the heat capacity would drop to zero. Any degree of freedom with energy spacings greater than k_BT are considered to be inactive  or frozen out; this is the situation for the vibrational degrees of freedom at room temperature and that is why the usual assumption is that they will not contribute. As we lower the temperature rotations will cease to contribute as well. The figure below presents an idealized plot of the heat capacity of a gas versus temperature.

Heat capacity of solids

We will consider Einsteins model for monatomic solids. Einstein proposed a simple model for such substances whereby the atoms only have vibrational energy (each atom can vibrate in three perpendicular directions around its lattice position). Specifically, the ‘Einstein Solid Model’ assumes that the atoms act like three-dimensional harmonic oscillators (with the vibrational motion of each atom in each perpendicular dimension entirely independent). This is equivalent to assuming that electronic states do not contribute so equation \(\ref{Qsolid}\) becomes:

\[Q(solid) = {q_{vib}^N}.\label{Q_einstein}\]

Near room temperature and for heavy atoms, such as most metals, the vibrations will be low frequency and thus "active". Following the same argument made for gases, this leads to just the last term in equation \(\ref{CV_ep1}\) and \(\ref{CVm_ep}\). As we are assuming three active vibrational degrees of freedom, we expect the molar heat capacity for solid pure elements near room temperature to be: 

\[C_{Vm} \approx 3R = 24.94 J mol^{-1} K^{-1}.\nonumber\]

In the early 1800s, two French scientists by the names of Pierre Louis Dulong and Alexis Therese Petit empirically discovered the same remarkable result. The Dulong-Petit Law is normally expressed in terms of the specific heat capacity (\(C_s\)) and the molar mass (\(M\)) of the metal

\[C_s M = C_{Vm} \approx 25 (J  K^{-1} mol^{-1}) \label{Dulong-Petit}\]

where \(C_s\) represents how much heat is required to raise the temperature of 'one gram' of that substance by one degree Kelvin. Dulong and Petit, as well as other scientists of their time, used this famous relationship as a means of establishing more accurate values for the atomic weight of metallic elements (by instead measuring the specific heat capacity of the element and using the Dulong-Petit relationship, which is a relatively simple method of establishing weights in comparison to the more disputable gravimetric methods that were being used at the time to establish the equivalent weights of elements).

Because only vibrations are involved it is relatively straightforward to derive the full temperature dependent expression for the heat capacity of the 'Einstein Solid Model' through statistical mechanics.  The average vibrational energy of a 1-D harmonic oscillator is:

\[\langle E_{vib} \rangle =\frac{ h\nu }{e^{h\nu/k_B T} -1}  =  \frac{  k_B \Theta_{vib} }{e^{\Theta_{v}/T} -1}. \label{Evib}\]

Taking the derivative with respect to temperature to get C_Vm and converting from per molecule to per mole we get:

\[ C_{V,m}^{1-D} = R \left( \dfrac{\Theta_v }{T} \right)^2 \left( \dfrac{e^{-\Theta_v/2T} }{1- e^{-\Theta_v/T}} \right) ^2 \label{1}\]

where \(R\) is the universal gas constant, \(T\) is absolute temperature, and \(Θ_v\) is called the ‘characteristic vibrational temperature’ of the oscillator, \(h\) represents Plank’s constant and \(k_B\) is Boltzmann’s constant.

Since the vibrations in each dimension are assumed to be independent, the expression for the constant volume molar heat capacity of a 'three-dimensional' Einstein Solid is obtained by simply multiplying Equation \ref{1} by three;

\[ C_{V,m}^{3-D} = 3R \left( \dfrac{\Theta_v }{T} \right)^2 \left( \dfrac{e^{-\Theta_v/2T} }{1- e^{-\Theta_v/T}} \right) ^2 \label{3}\]

The temperature variation of the heat capacity of most metallic solids is well described by equation \ref{3}. Furthermore, plots of equation \ref{3} as a function of temperature for metals with widely varying vibrational frequencies reveal that the heat capacity always approaches the same asymptotic limit of \(3R\) at high temperatures. Stated another way, at high temperatures

\[ \lim_{T \to \infty} \left[ \left( \dfrac{\Theta_v }{T} \right)^2 \left( \dfrac{e^{-\Theta_v/2T} }{1- e^{-\Theta_v/T}} \right) ^2 \right] = 1 \label{4}\]

and Equation \ref{3} reduces to the value we got previously:

\[ \lim_{T \to \infty} \left[ C_{V,m}^{3D} \right] = 3R \label{5}\]

In addition to predicting the high temperature limit correctly the model also predicts correctly that the heat capacity will approach zero as \(T \to 0 K\). The denominator in equation \(\ref{3}\) approaches zero more slowly than the numerator.

Exercises

Consult the CRC Handbook of Chemistry and Physics (CRC Press: Boca Raton, FL) or other reliable source of data and compile a table of specific heat capacities for a large number of elements that are known to exist as monoatomic solids at room temperature. Also look up and record the molar mass of these elements. The elements that you consider should be restricted to those appearing in groups 1-14 of the periodic table. Make sure you generate a fairly large list which includes a number of elements that are normally considered as metallic in character (such as copper, iron, sodium, lithium, gold, platinum, barium, and aluminum), but also some non-metallic elements that are nonetheless monoatomic isotropic solids (such as carbon-diamond, beryllium, boron, and silicon). Heat capacities that are usually reported in the literature are not actual constant volume heat capacities (\(C_v\)), but are instead constant pressure heat capacities (\(C_p\)). Fortunately, \(C_p\) and \(C_v\) are essentially equal for simple solids (within the level of precision that we consider in this exercise), and you can assume that the values from the CRC Handbook represent \(C_s\).

1. Enter the element name, the specific heat capacity, and the molar mass of each element in a your favorite data analysis tool. Calculate the product of specific heat and molar mass for each element and calculate how much this product differs from the Dulong-Petit prediction (express your result as a percent difference relative to \(3R\)).
2. Assess the generality of the Dulong-Petit law in an alternate way by generating a plot of specific heat as a function of reciprocal Molar Mass (\(C_s\) versus \(1/M\)), which should be linear with a slope equal to 3R if the data behave according to Equation \ref{Dulong-Petit}.
3. Inspect your results from 1 and 2 above and identify any elements that significantly deviate from the Dulong-Petit law. When they occur, do deviations tend to be smaller or larger than 3R? Does the degree of deviation from the Dulong-Petit law seem to correlate with periodic trends in metallic (or covalent) bonding for these elements? Do deviations tend to occur more readily for elements of smaller or higher atomic weight? Explain how the type of bonding and the magnitude of the atomic weight can lead to deviations from the arguments made in Equations \ref{Dulong-Petit}-\ref{5} above.
4. Use the plotting method that you employed in step 2 above as a means of determining a value for the universal gas constant (\(R\)) - but make sure you throw-out any specific heat data for elements that you suspect do not fall within the limit \(Θ_v \ll 298 \,K\). Calculate the percent error in the value of \(R\) that you determine.
5. Verify that the limit expressed in Equation \ref{4} above is true (HINT: expand each of the exponential terms in a power series and note that higher-order terms are negligible in the limit \(T \gg Θ_v\)).