Problem 6 - Chemistry and Chemical Reactivity

Question

The specific heat capacity of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is \(1.74 \mathrm{J} / \mathrm{g} \cdot \mathrm{K} .\) What is its molar heat capacity (in \(\mathrm{J} / \mathrm{mol} \cdot \mathrm{K}) ?\)

Step-by-Step Solution

Answer

The molar heat capacity of benzene is 135.93 J/mol·K.

Step 1: Understand the Problem

We are given the specific heat capacity of benzene and need to find the molar heat capacity. Specific heat capacity is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 K, while molar heat capacity refers to the heat required to raise the temperature of 1 mole by 1 K.

Step 2: Use the Formula

The formula to calculate molar heat capacity from specific heat capacity is: \[molar\ heat\ capacity = specific\ heat\ capacity \times molar\ mass\]

Step 3: Find the Molar Mass of Benzene

Benzene's chemical formula is \(\mathrm{C}_{6} \mathrm{H}_{6}\). Calculate the molar mass by summing the atomic masses of all the atoms in a molecule: - Carbon (C): 12.01 g/mol- Hydrogen (H): 1.01 g/molThus, \[Molar\ mass = (6 \times 12.01) + (6 \times 1.01) = 78.12\ g/mol\]

Step 4: Calculate the Molar Heat Capacity

Substitute the known values into the formula from Step 2: \[Molar\ heat\ capacity = 1.74\ \mathrm{J/g\cdot K} \times 78.12\ \mathrm{g/mol} = 135.93\ \mathrm{J/mol\cdot K}\]

Step 5: Conclusion

We have found that the molar heat capacity of benzene is \(135.93\ \mathrm{J/mol \cdot K}\).

Key Concepts

Specific Heat Capacity

Specific heat capacity is a property of a material that describes how much energy is required to increase the temperature of a single gram by one degree Kelvin (K). In simpler terms, it's a measure of how resistant a substance is to changing in temperature. For benzene, this value is given as \(1.74\, \text{J/g} \cdot \text{K}\). This means 1.74 joules of energy are needed to raise the temperature of 1 gram of benzene by 1 K.
Understanding this concept is crucial in fields like chemistry and physics because it helps to predict how substances will respond when energy is added. This property is unique for each material due to differences in molecular structure and bonding.

Molar Mass

Molar mass is the mass of one mole of a substance. It is usually expressed in grams per mole (g/mol). Knowing the molar mass is essential when converting between mass and moles, which are used frequently in chemical calculations.
For benzene \((\text{C}_6\text{H}_6)\), we compute the molar mass by summing the atomic masses of its constituent atoms.

• Carbon \((\text{C})\): 12.01 g/mol, with 6 atoms contributing a total of \(6 \times 12.01 = 72.06\) g/mol
• Hydrogen \((\text{H})\): 1.01 g/mol, with 6 atoms contributing a total of \(6 \times 1.01 = 6.06\) g/mol

Adding these gives benzene a molar mass of \(78.12\, \text{g/mol}\). This value is vital for determining how much heat is needed per mole, leading us to calculate the molar heat capacity.

Chemical Formula

The chemical formula of a compound reveals the types and numbers of atoms present in a molecule. Benzene's chemical formula is \(\text{C}_6\text{H}_6\), indicating it contains six carbon (C) atoms and six hydrogen (H) atoms.
Understanding chemical formulas is fundamental for multiple reasons:

• They allow us to determine the molar mass, essential for many chemical equations and reactions.
• They inform about the molecular structure and the chemical behavior of a substance since the formula reflects specific bonding patterns.

This aids in visualizing how the compound might interact under different conditions, such as heating.

Thermal Properties of Benzene

Benzene's thermal properties, such as its specific and molar heat capacity, are significant in understanding its behavior under thermal conditions. The molar heat capacity of benzene, calculated from its specific heat capacity and molar mass, is \(135.93\, \text{J/mol} \cdot \text{K}\).
The calculation is performed using the relationship: \[\text{Molar Heat Capacity} = \text{Specific Heat Capacity} \times \text{Molar Mass}\]By substituting benzene's specific heat capacity of \(1.74\, \text{J/g} \cdot \text{K}\) and its molar mass of \(78.12\, \text{g/mol}\), we derive the given molar heat capacity.
This knowledge helps predict how benzene will absorb and transfer heat. It's essential for applications where temperature control of benzene-containing systems is required, such as in chemical manufacturing and environmental regulation.