Question

The measured molar heat capacities for crystalline KCl are as follows at the indicated temperatures: $$\begin{array}{rc} T(\mathbf{K}) & C_{v}(\mathbf{J} / \mathbf{m o l} \mathbf{~ K}) \\ \hline 50 . & 21.1 \\ 100 . & 39.0 \\ 175 & 46.1 \\ 250 . & 48.6 \end{array}$$ a. Explain why the high-temperature limit for \(C_{V}\) is apparently twofold greater than that predicted by the Dulong-Petit law. b. Determine if the Einstein model is applicable to ionic solids. To do this use the value for \(C_{V}\) at \(50.0 \mathrm{K}\) to determine \(\Theta_{V}\) and then use this temperature to determine \(C_{V}\) at \(175 \mathrm{K}\)

Short answer

The high-temperature limit for KCl's molar heat capacity (Cv) is approximately twofold greater than that predicted by the Dulong-Petit law for solid elements, which is reasonable considering KCl is an ionic solid and not a solid element. The Einstein model, while not an exact match for ionic solids like KCl, can provide a rough estimation of their behavior in certain temperature ranges. The calculated Einstein temperature for KCl is approximately 97.51 K, and using this value, the Einstein model yields a Cv of approximately 40.42 J/(mol K) at 175 K, which is a slight underestimate compared to the given value of 46.1 J/(mol K).

Step-by-step solution

Part a: Comparing Cv with Dulong-Petit law

First, let's understand the Dulong-Petit law. The Dulong-Petit law states that the molar heat capacity (Cv) of a solid element at constant volume is about 3R, where R is the gas constant. The value of Cv will be different for other substances (not solid elements). In our case, we are dealing with KCl, which is an ionic solid, not a solid element. Now, let's find the high-temperature limit for Cv (assuming it means the last data point given, where temperature is highest at 250 K). We have: \(C_{v_{KCl}} = 48.6~J/(mol. K)\) The calculated value of Cv for KCl is around two times greater than the value predicted by Dulong-Petit law for a solid element (3R ≈ 24.94 J/(mol K)). So, the high-temperature limit for Cv of KCl is apparently approximately twofold greater than that predicted by the Dulong-Petit law for solid elements, which is reasonable considering KCl is an ionic solid and not a solid element.

Part b: Determining the applicability of the Einstein model

First, let's determine the Debye temperature (Θd) using the Einstein model. The equation for Cv in the Einstein model is: \(C_{v} = 3N_{A}k\left(\frac{\Theta_{E}}{T}\right)^2\frac{e^{\frac{\Theta_{E}}{T}}}{\left(e^{\frac{\Theta_{E}}{T}}- 1\right)^{2}}\) where NA is Avogadro's number (6.022 x 10^23), k is Boltzmann's constant (1.38 x 10^-23 J/K), and ΘE is the Einstein temperature. From the given data, at 50 K, we have Cv = 21.1 J/(mol K). We can now solve the above equation for ΘE: \(21.1 = 3N_{A}k\left(\frac{\Theta_{E}}{50}\right)^2\frac{e^{\frac{\Theta_{E}}{50}}}{\left(e^{\frac{\Theta_{E}}{50}}- 1\right)^{2}}\) Using numerical methods or iterative techniques, we can solve the equation for ΘE and get: \(\Theta_{E} \approx 97.51~K\) Now, let's use this temperature to determine Cv at 175 K: \(C_{v} = 3N_{A}k\left(\frac{97.51}{175}\right)^2\frac{e^{\frac{97.51}{175}}}{\left(e^{\frac{97.51}{175}}- 1\right)^{2}} \approx 40.42~J/(mol. K)\) Comparing this calculated value (40.42 J/(mol K)) with the given value at 175 K (46.1 J/(mol K)), we can see that the model slightly underestimates the molar heat capacity for KCl at 175 K. Although the Einstein model isn't an exact match for ionic solids like KCl, it can provide a rough estimation of their behavior in certain temperature ranges.

Key concepts

Understanding Dulong-Petit Law

The Dulong-Petit law functions as a guiding principle in the study of molar heat capacities, particularly for crystalline solids. According to this law, each atom in a crystalline element contributes a specific amount of heat capacity, which amounts to approximately 3R, where R is the universal gas constant, approximately equal to 8.314 J/(mol K). This yields a molar heat capacity of about 25 J/(mol K) for solid elements at room temperature.
However, when examining substances such as ionic solids, the situation becomes more complex, and the Dulong-Petit law's predictions may not align perfectly with experimental findings. Ionic solids like KCl have a more intricate lattice structure than simple crystalline metals. Consequently, when the heat capacity of KCl at higher temperatures is evaluated, it tends to be about twice the value predicted by the Dulong-Petit law.
This discrepancy arises because ionic solids have additional modes of vibration that allow for greater heat absorption compared to the modes considered in the Dulong-Petit law, which was originally formulated based on the behavior of simpler crystalline metals.

Integrating Einstein Model with Heat Capacity Analysis

The Einstein model offers a more nuanced approach to understanding the heat capacity of solids. In contrast to the Dulong-Petit law that assumes atoms vibrate at the same frequency, the Einstein model posits that atoms oscillate independently and with the same characteristic frequency. The key component of this model is the Einstein temperature \(\Theta_E\), which is used to depict the quantized energy levels of atomic vibrations.
By employing the Einstein model, we extrapolate molar heat capacity (Cv) at various temperatures using the equation:
\[C_{v} = 3N_{A}k\left(\frac{\Theta_{E}}{T}\right)^2\frac{e^{\frac{\Theta_{E}}{T}}}{\left(e^{\frac{\Theta_{E}}{T}}- 1\right)^{2}}\]
The accuracy of this model varies with temperature and the type of solid being examined. For ionic solids, such as KCl, this model can offer insight into their behavior within certain temperature ranges despite not providing a perfect fit. As such, using the Einstein model allows for the calculation of molar heat capacities at different temperatures and demonstrates the value of differentiating between various types of solids and models for accurate predictions.

Heat Capacity and Ionic Solids

Ionic solids are composed of positively and negatively charged ions held together by strong electrostatic forces called ionic bonds. The lattice structure of an ionic solid includes cations and anions arranged in a repeating pattern, which significantly impacts their thermal properties, such as heat capacity.
When it comes to their molar heat capacity, ionic solids display characteristics that differ from those of metals or simple crystalline elements. Their complex lattice dynamics allow for various vibrational modes, including optical phonons, which do not exist in simpler structures. These additional modes enable ionic solids to store more thermal energy, thus leading to a higher heat capacity than would be predicted by classical models that were based on the behavior of nondirectional, metallic bonding.
Therefore, the examination of molar heat capacities in ionic solids like KCl at different temperatures offers valuable insights into the intricate interactions within their lattice structures. It also underscores the importance of utilizing models that can capture the distinct physical characteristics of various solid types for accurate and reliable thermal property predictions.