Problem 5 - Chemistry and Chemical Reactivity

Question

The molar heat capacity of mercury is \(28.1 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) What is the specific heat capacity of this metal in \(\mathrm{J} / \mathrm{g} \cdot \mathrm{K}\) ?

Step-by-Step Solution

Answer

The specific heat capacity of mercury is approximately \( 0.140 \, \mathrm{J/g} \cdot \mathrm{K} \).

Step 1: Understand the Relationship

The molar heat capacity, denoted as \( C_m \), is given as \( 28.1 \, \mathrm{J} / \mathrm{mol} \cdot \mathrm{K} \). Specific heat capacity, denoted as \( c \), is related to the molar heat capacity by the formula: \[ c = \frac{C_m}{M} \] where \( M \) is the molar mass of the substance in \( \mathrm{g/mol} \).

Step 2: Determine the Molar Mass of Mercury

The molar mass of mercury (Hg) is approximately \( 200.59 \, \mathrm{g/mol} \). This value is retrieved from the periodic table, which lists the average atomic masses of elements.

Step 3: Apply the Formula

Substitute \( C_m = 28.1 \, \mathrm{J/mol} \cdot \mathrm{K} \) and \( M = 200.59 \, \mathrm{g/mol} \) into the formula: \[ c = \frac{28.1}{200.59} \] This will give us the specific heat capacity in \( \mathrm{J/g} \cdot \mathrm{K} \).

Step 4: Perform the Calculation

Calculating the specific heat capacity: \( c = \frac{28.1}{200.59} \approx 0.140 \mathrm{J/g} \cdot \mathrm{K} \). This represents the amount of heat needed to raise 1 gram of mercury by 1 Kelvin.

Key Concepts

Molar Heat Capacity

Molar heat capacity is a fundamental concept in thermodynamics. It represents the amount of heat required to raise the temperature of one mole of a substance by one Kelvin. In our case, the molar heat capacity of mercury is given as 28.1 J/mol·K. This value shows us how much energy is needed per mole for thermal changes.
Understanding molar heat capacity helps in predicting how substances will respond to heat. Depending on the substance, molar heat capacity can vary significantly, influencing its use in thermal applications. Additionally, this thermodynamic quantity is crucial for various calculations in chemistry and physics, providing insights into the energetic behavior of materials.
Remember, when working with molar heat capacities, it’s often beneficial to have a periodic table nearby to determine the necessary molar masses for your calculations.

Molar Mass of Elements

The molar mass of an element is the weight of one mole of atoms of that element, usually expressed in grams per mole (g/mol). For mercury, its molar mass is approximately 200.59 g/mol, as seen in the periodic table.
The molar mass not only helps in converting moles to grams but also plays a critical role in various chemical calculations, including those involving specific heat capacity. In our exercise, the molar mass of mercury is essential for converting its molar heat capacity to its specific heat capacity.
To find the molar mass of an element:

• Consult the periodic table
• Observe the atomic mass listed below the symbol of the element
• Round to the nearest whole number when necessary for simpler calculations

Having a strong grasp of molar mass concepts allows for more straightforward and accurate computations in chemistry.

Thermodynamic Calculations

Thermodynamic calculations are essential for understanding energy changes in systems. In this context, converting molar heat capacity to specific heat capacity is a common type of calculation. Using the formula: \[ c = \frac{C_m}{M} \]where \(C_m\) is the molar heat capacity and \(M\) is the molar mass, we can determine how much energy is needed to raise the temperature of one gram of a substance by one Kelvin.
This approach aids in practical applications like designing heating and cooling systems, understanding natural processes, and studying materials' thermal properties.
To perform thermodynamic calculations effectively:

• Ensure your units are consistent. Mix-ups can lead to incorrect conclusions.
• Double-check that all values, especially molar mass and heat capacity, are accurate.
• Use LaTeX or clear notation to handle mathematical expressions, enhancing the clarity of your presentation.

Mastery of these calculations opens the door to more complex studies in material science and engineering.