One mole of \(\mathrm{H}_{2} \mathrm{O}(g)\) at \(1.00 \mathrm{~atm}\) and \(100 .^{\circ} \mathrm{C}\) occupies a volume of \(30.6 \mathrm{~L}\). When one mole of \(\mathrm{H}_{2} \mathrm{O}(g)\) is condensed to one mole of \(\mathrm{H}_{2} \mathrm{O}(l)\) at \(1.00 \mathrm{~atm}\) and \(100 .{ }^{\circ} \mathrm{C}, 40.66 \mathrm{~kJ}\) of heat is released. If the density of \(\mathrm{H}_{2} \mathrm{O}(l)\) at this temperature and pressure is \(0.996 \mathrm{~g} / \mathrm{cm}^{3}\), calculate \(\Delta E\) for the condensation of one mole of water at \(1.00 \mathrm{~atm}\) and \(100 .{ }^{\circ} \mathrm{C}\).

Condensation is a fundamental physical process in which water vapor in the atmosphere changes into liquid water. It's the reverse of vaporization, where liquid water becomes water vapor. Understanding condensation is key for grasping other important concepts in chemistry and thermodynamics, such as phase transitions and energy exchange.

When water vapor condenses, it releases energy in the form of heat. This release occurs because the gaseous state, which requires more energy due to the higher kinetic energy of the particles, transitions into a liquid state with lower kinetic energy. The conservation of energy must account for this energy difference, typically by releasing heat to the surroundings. That's why you might observe water droplets forming on the side of a cold beverage can — the moisture from the air condenses upon contacting the colder surface.

Consider the condensation of one mole of water vapor: it involves a decrease in volume as the gas particles become more ordered as a liquid. The textbook exercise asks the student to calculate the change in internal energy when water vapor condenses at a specific temperature and pressure, illustrating the transfer of energy that occurs during the phase change.

In the context of chemistry and physics, internal energy refers to the total energy contained within a system, comprising both kinetic and potential energies of particles. Internal energy change, denoted as \( \Delta E \), is a crucial concept in the study of thermodynamics. It is the sum of all the energy transformations that occur within a system, excluding any work done by or on the system.

In the process of condensation, the internal energy of the system decreases due to the release of heat. Applying the First Law of Thermodynamics, we know that the change in internal energy of a system is the heat added to it minus the work done by it. That is:
\[\Delta E = q - w\]
Where \( q \) is the heat exchanged and \( w \) is the work done on or by the system. In this case, \( q \) is negative because the system is releasing heat, indicating an exothermic process. Understanding how to manipulate this formula and account for the variables involved is critical for solving problems in thermodynamics, as seen in the provided exercise.

The enthalpy of vaporization, commonly notated as \( \Delta H_{vap} \), is the amount of energy required for one mole of a substance to transform from its liquid phase into its vapor phase without a change in temperature. This energy is needed to overcome the intermolecular forces present in the liquid state, which is conversely released when a substance transitions from vapor to liquid, a process known as condensation.

For example, when water vapor condenses to liquid water, it releases energy equivalent to its enthalpy of vaporization. This is observed when liquid water forms from the condensation of steam. The same amount of energy released during condensation is required to turn that liquid water back into water vapor.

The exercise highlights the use of the released heat in the calculation of internal energy change during the condensation process. The enthalpy of vaporization is vital for calculating the heat involved (\( q \)) when the pressure and temperature are constant, such as in the case of the exercise, and it's the conceptual opposite of the internal energy change during condensation.