91Ó°ÊÓ

Enthalpy of vaporization, often denoted as \( \Delta H_{vap} \), is a crucial concept in the study of phase changes, particularly when a substance transitions from its liquid phase to its vapor phase at the boiling point. It represents the amount of energy required to vaporize one mole of a substance at constant pressure without any change in temperature.

In the example provided, the enthalpy of vaporization is given as \(30.7\) kJ/mol at \(80\) degrees Celsius. This value indicates the heat required to convert one mole of the liquid into gas form at its boiling point under the provided conditions. It's important to note that while the enthalpy of vaporization is a measure of the heat absorbed during the process, it differs from the total change in internal energy, which also accounts for work done on or by the system.

To understand this concept fully, students should recognize the use of the enthalpy of vaporization in calculating the heat absorbed during phase transitions, particularly at constant pressure. This ties directly into the first law of thermodynamics, with the heat absorbed being a component of the energy changes within the system.

The first law of thermodynamics, also known as the law of energy conservation, states that the total energy of an isolated system is constant. Energy can be transformed from one form to another, but it cannot be created or destroyed.

In the context of our exercise, the first law can be expressed as \( \Delta E = q + w \), where \( \Delta E \) represents the change in internal energy of the system, \( q \) is the heat added to the system, and \( w \) is the work done by the system against the external environment. When a substance undergoes vaporization, it absorbs heat and potentially does work, expanding against atmospheric pressure.

The detailed solution to the exercise demonstrates the application of the first law by using the given enthalpy of vaporization (which equates to \( q \) at constant pressure) and the calculated work done during the vaporization to determine the change in internal energy of the molten liquid. This approach underlines the intertwined nature of heat and work in thermodynamic processes and the importance of considering both when calculating energy changes.

The Ideal Gas Law is a fundamental equation in chemistry and physics which connects the pressure \( P \), volume \( V \), temperature \( T \), and the amount of gas in moles \( n \), along with the ideal gas constant \( R \). The law is usually stated as:

\[ PV = nRT \]
This relationship is critical when studying gases and their behavior under different conditions of pressure, temperature, and volume. In the context of our problem, it allows us to calculate the volume of a gas when it vaporizes from its liquid state. Given that we’re dealing with the vaporization process of one mole of liquid, the change in volume \( \Delta V \) can be accurately predicted using the Ideal Gas Law, assuming that the behavior of the vapor is ideally following the gas laws.

To adhere to the Ideal Gas Law, the solution uses the known values of temperature, the ideal gas constant, and the number of moles of the vapor. It assumes that the volume of the liquid is negligible, simplifying the calculation to determine the volume of gas produced during vaporization, which in turn is used to calculate the work done during the phase change.