The standard entropy of vaporization of benzene is approximately \(85 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\) at its boiling point. (a) Estimate the standard enthalpy of vaporization of benzene at its normal boiling point of \(80 .^{\circ} \mathrm{C}\). (b) What is the standard entropy change of the surroundings when \(10 . \mathrm{g}\) of benzene, \(\mathrm{C}_{6} \mathrm{H}_{6}\), vaporizes at its normal boiling point?

The Gibbs free energy equation is fundamental in thermodynamics and describes the total amount of usable energy called 'free' energy, within a chemical system. Given by the formula \( \Delta G = \Delta H - T\Delta S \), it serves as a critical tool in predicting the direction of a chemical reaction under constant temperature and pressure.

The equation combines the concepts of enthalpy (\(\Delta H\)), which denotes the heat content or total energy, and entropy (\(\Delta S\)), indicative of the disorder or randomness of the system. Temperature (\(T\)) in Kelvin degrees acts as a pivot to balance between enthalpy and entropy. At equilibrium, especially during phase changes like vaporization at the boiling point, Gibbs free energy change \(\Delta G\) is zero. This is because the process occurs spontaneously and no additional energy is required to drive it forward. By rearranging the equation to solve for \(\Delta H_{vap}\) when \(\Delta G = 0\) at the boiling point, we obtain an elegant approach to estimate the enthalpy of vaporization — a measure of energy required to convert a substance from liquid to gaseous state.

Enthalpy of vaporization, often denoted as \(\Delta H_{vap}\), quantifies the heat energy needed to vaporize a specific amount of a substance at its boiling point under standard pressure. This physical property is essential because it provides insight into the strength of intermolecular forces within a liquid – the stronger these forces, the higher the enthalpy of vaporization.

The calculation in our textbook problem cleverly employs the condition where Gibbs free energy is zero at the boiling point. The enthalpy of vaporization is then directly proportional to the entropy of vaporization and the absolute temperature. It's noteworthy that enthalpies of vaporization are typically expressed in Joules per mole (J/mol) or kilojoules per mole (kJ/mol), providing a clear understanding of the energy required on a per-mole basis.

Molar mass refers to the mass of one mole of a substance and is a crucial concept for converting between mass and number of moles in chemistry. The molar mass of a compound like benzene, \(C_6H_6\), is calculated by summing the molar masses of its constituent atoms, carbon and hydrogen, which are determined from the periodic table.

Specifically, the molar mass aids in quantifying how much of a substance is present. This calculation is fundamental in stoichiometry, the part of chemistry that deals with the quantities of reactants and products in chemical reactions. In practical terms, knowing the molar mass allows a student to determine the number of molecules in a given mass of a compound – a critical step in many quantitative exercises like the one illustrated here.

The concept of entropy extends beyond the system in question to include the surroundings – essentially, the rest of the universe. In a thermodynamic process, the total entropy change of the universe equals the sum of the changes in entropy of the system and its surroundings. When a substance like benzene vaporizes, the system absorbs heat, increasing its own entropy. The Second Law of Thermodynamics ensures that this is compensated by a greater entropy increase in the surroundings, to favor the spontaneous process.

The entropy change for the surroundings during vaporization is calculated negatively from the system's entropy change (because energy is taken from the surroundings) and normalized by the temperature. This key step highlights a universal balance: while a system may become more ordered by losing entropy, its surroundings must become more disordered, maintaining the natural trend towards increased overall disorder or entropy in the universe.