Calculate the standard entropy of vaporization of water at \(85^{\circ} \mathrm{C}\), given that its standard entropy of vaporization at \(100 .{ }^{\circ} \mathrm{C}\) is \(109.0 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\) and the molar heat capacities at constant pressure of liquid water and water vapor are \(75.3 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\) and \(33.6 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\), respectively, in this range.

The Clausius-Clapeyron equation is a fundamental relationship in thermodynamics that describes the phase transition between two states of matter, such as liquid and vapor. It helps us understand how the pressure and temperature of a substance are related during a phase change.

Understanding this equation is crucial when considering the entropy of vaporization, which is a measure of the energy dispersal during the process of converting a liquid to a gas. The Clausius-Clapeyron equation specifically links the temperature, pressure, and the enthalpy of vaporization (∆H_vap) and provides insight into how entropy changes with temperature.

For practical purposes, it's important to remember that while the Clausius-Clapeyron equation can appear formidable, it essentially allows us to calculate important thermodynamic quantities that are otherwise difficult to measure directly, such as changes in enthalpy or entropy for phase transitions under varying atmospheric conditions.

Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy transfer and conversion between them. In the context of entropy of vaporization, thermodynamics helps us understand the energy changes associated with the phase change of a substance.

One of the primary concepts of thermodynamics is the first law, which states that energy cannot be created or destroyed, only transformed. When a substance vaporizes, energy is absorbed to break intermolecular forces, and this energy is accounted for in the enthalpy of vaporization. The second law of thermodynamics introduces the notion of entropy, a measure of disorder or randomness in a system. During vaporization, the entropy increases significantly due to the greater freedom and disorder of the molecules in the gaseous phase compared to the liquid phase.

Overall, thermodynamics provides a framework for predicting and understanding how substances behave during phase transitions, like vaporization, related to heat, energy, and entropy.

Heat capacity is a material-specific property that indicates the amount of heat required to raise the temperature of a given substance by one degree Celsius (or one Kelvin). Specifically, the molar heat capacity at constant pressure (C_P) describes how much heat energy is needed per mole of a substance to increase its temperature under constant pressure conditions.

During the process of heating a substance from one temperature to another, the heat capacity plays a key role in determining the total energy change – and therefore the change in entropy – of the substance. Heat capacity can vary with temperature, but for the calculation of entropy changes over small temperature ranges, such as in the exercise given, it is often assumed to be constant.

Understanding the concept of heat capacity is essential for predicting how a substance's temperature will change in response to energy input, which in turn affects the entropy of a system during thermal processes like vaporization.

A phase change is a physical process in which a substance transitions from one state of matter – solid, liquid, or gas – to another. Each phase has particular characteristics and properties that distinguish it from the others. The phase change of water from liquid to vapor, also known as vaporization or boiling, involves significant energy changes within the substance.

During the phase change, the added energy doesn't raise the temperature but instead works to transform the physical state by breaking intermolecular bonds. In thermodynamics, the entropy change associated with a phase change is a key concept. When a liquid vaporizes, the molecules spread out, increasing the disorder of the system, and therefore the entropy.

Understanding phase changes is fundamental for interpreting phenomena in many scientific and engineering fields, such as meteorology, environmental science, and mechanical engineering, where the managing of energies and thermodynamic processes is crucial.