Question

According to the law of Dulong and Petit the specific heats of solids should differ only because of differences in \(M_{a}\). Calculate \(M_{a}\) and \(c\) for \(\mathrm{MgSiO}_{3}\) and \(\mathrm{MgO} .\) The measured values of \(c\) at standard conditions of temperature and pressure are 815 \(\mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}\) for \(\mathrm{MgSiO}_{3}\) and \(924 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\) for \(\mathrm{MgO} .\) A MATLAB solution to this problem is provided in Appendix \(D\).

Short answer

Molar masses are 100.39 g/mol for MgSiO3 and 40.31 g/mol for MgO. Specific heats are 81.9 J/mol⋅K for MgSiO3 and 37.25 J/mol⋅K for MgO.

Step-by-step solution

Understanding Dulong and Petit's Law

The law of Dulong and Petit states that the molar specific heat capacity (C) of a solid element is approximately 3R, where R is the universal gas constant ( R = 8.314 J/(mol⋅K)). For a compound, the specific heat capacity can be found using the formula: \[ C = \frac{3R}{M_a} \] where \(M_a\) is the molar mass of the compound.

Calculate Molar Mass (MgSiO3)

To find the molar mass \(M_a\) of \(\mathrm{MgSiO}_{3}\), sum the atomic masses of its constituent atoms: - Mg: 24.31 g/mol - Si: 28.09 g/mol - O (3 atoms): 16.00 g/mol each\[ M_a (\mathrm{MgSiO}_{3}) = 24.31 + 28.09 + 3 \times 16.00 = 100.39 \, \text{g/mol} \]

Calculate Molar Mass (MgO)

To find the molar mass \(M_a\) of \(\mathrm{MgO}\), sum the atomic masses of its constituent atoms:- Mg: 24.31 g/mol - O: 16.00 g/mol \[ M_a (\mathrm{MgO}) = 24.31 + 16.00 = 40.31 \, \text{g/mol} \]

Calculate Specific Heat Capacity (MgSiO3)

The measured specific heat \( c \) for \( \mathrm{MgSiO}_{3} \) is given as 815 J/(kg⋅K). Convert it to molar specific heat using its molar mass: \[ C_{MgSiO_3} = 815 \, \mathrm{J} / \mathrm{kg} \, \mathrm{K} \times \frac{100.39 \, \text{g/mol}}{1000 \, \text{g/kg}} = 81.9 \, \mathrm{J/mol\, K} \]

Calculate Specific Heat Capacity (MgO)

The measured specific heat \( c \) for \( \mathrm{MgO} \) is given as 924 J/(kg⋅K). Convert this to molar specific heat using its molar mass: \[ C_{MgO} = 924 \, \mathrm{J} / \mathrm{kg} \, \mathrm{K} \times \frac{40.31 \, \text{g/mol}}{1000 \, \text{g/kg}} = 37.25 \, \mathrm{J/mol\, K} \]

Analyze the Results

Comparing the calculated molar heat capacities to Dulong and Petit's value of 3R (approximately 24.9 J/mol K), the calculated values of 81.9 J/mol K for \(\mathrm{MgSiO}_{3}\) and 37.25 J/mol K for \(\mathrm{MgO}\) deviate from this value due to the complexity of the compounds not adhering strictly to the rule applicable for simple solids.

Key concepts

Dulong and Petit's Law

Dulong and Petit's Law is a crucial principle in thermodynamics which states that the molar specific heat capacity of many solid elements is approximately equal to 3R, where R is the universal gas constant with a value of 8.314 J/(mol⋅K). This law provides a way of comparing and predicting the specific heat capacities of elements and is often used for making estimations in the study of thermodynamics.
This empirical law holds well for many metals at room temperature but is less accurate for compounds and nonmetals. In practice, the deviations that occur are due to various factors, including atomic structure and bonding. Therefore, it becomes essential to consider that while Dulong and Petit's Law offers a good approximation for simple atoms, the complexity of compounds may require different considerations.
Understanding this law helps in the exploration of how different chemical compounds interact with heat, making it hugely significant in fields like material science and physical chemistry.

Molar Mass Calculation

Calculating the molar mass of a compound is a fundamental concept in chemistry. It involves summing up the atomic masses of all atoms present in a molecule. For example, to calculate the molar mass of magnesium silicate (MgSiO₃), one must add the atomic masses of magnesium (Mg), silicon (Si), and three oxygen (O) atoms:

• Magnesium (Mg): 24.31 g/mol
• Silicon (Si): 28.09 g/mol
• Oxygen (O): 16.00 g/mol times 3

Performing the calculation, we find:\[ M_a (MgSiO₃) = 24.31 + 28.09 + 3 \times 16.00 = 100.39 \, \text{g/mol} \]
Similarly, for magnesium oxide (MgO), the calculation is straightforward, adding the atomic masses of magnesium and oxygen:

• Magnesium (Mg): 24.31 g/mol
• Oxygen (O): 16.00 g/mol

Thus:\[ M_a (MgO) = 24.31 + 16.00 = 40.31 \, \text{g/mol} \]
Understanding molar mass is critical, as it allows chemists to convert between grams and moles, aiding in the calculation of chemical reactions and properties.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy. In connection to Dulong and Petit's Law, thermodynamics helps us understand how substances absorb heat and how they respond to changes in temperature. It encompasses various laws and principles that describe how energy moves within a system and is essential in explaining both the microscopic and macroscopic physical processes.
The concepts of energy conservation (first law), entropy (second law), and absolute zero (third law) provide a solid framework for comprehending many processes, including those involving heat capacity. Heat capacity, specifically, is an intrinsic property that shows a material's ability to hold on and transfer heat.
In addition to helping with calculations for molar specific heat capacities as seen with Dulong and Petit's Law, thermodynamics plays a pivotal role in procedures ranging from designing engines to understanding natural phenomena. Grasping these concepts is key for anyone studying or working in fields that involve heat transfer and energy conversion.

Specific Heat Capacity

Specific heat capacity is a vital concept in thermodynamics and helps determine how much heat energy a substance can store. It is defined as the amount of heat required to raise the temperature of one kilogram of a substance by one Kelvin. In simple terms, it measures how much heat a substance can take.
For instance, the specific heat capacities provided in the original problem for magnesium silicate (MgSiO₃) and magnesium oxide (MgO) are 815 J/kg⋅K and 924 J/kg⋅K, respectively. These values indicate the heat needed to change their temperatures and are necessary to convert into molar specific heat capacities using their respective molar masses. This conversion helps in understanding the nature of the compound and in comparing it to Dulong and Petit's prediction.
Understanding specific heat capacity allows scientists and engineers to predict how different materials will react under various temperature conditions, which is useful in applications such as thermal energy storage and heat exchangers.

Chemical Compounds

Chemical compounds are substances formed by two or more different elements bonded together. Each compound has unique properties, including melting points, boiling points, and specific heat capacities, which depend on the nature of the bonding and molecular structure. Take magnesium silicate (MgSiO₃) and magnesium oxide (MgO), for instance; these compounds have different chemical behaviors and heat capacities.
The composition of these compounds contributes to the various differences in properties observed. For example, complex structures have more intricate interactions, affecting their heat capacity and response to temperature changes.
Studying chemical compounds allows us to understand their role in various reactions and their significance in scientific and industrial applications. Comprehending their specific heat capacities, as part of their thermodynamic properties, aids in the development of new materials with desired thermal properties.