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The first law of thermodynamics, often called the law of energy conservation, establishes a fundamental principle crucial to the science of physics and chemistry. It states that the energy of a closed system is constant; energy can neither be created nor destroyed, but only transformed from one form to another. In the exercise, this law is applied to the phase change of water from liquid to gas.

When water vaporizes, the energy used in the process goes into changing the internal energy of the water and doing work against external pressure, such as the atmosphere. This is a direct application of the first law, as the molar enthalpy of vaporization is the energy input (heat absorbed, q) required to change the state of one mole of water without changing its temperature. Understanding this concept is essential for explaining why two different fractions of energy are calculated in the solution—the internal energy change and the work done on the surroundings.

Enthalpy change, denoted as ΔH, is a measure of the total heat content change in a system at constant pressure. It is a fundamental concept in thermodynamics and is particularly significant when analyzing phase changes, such as vaporization. In the given exercise, the molar enthalpy of vaporization refers to the amount of heat, ΔH, needed to turn one mole of liquid water into water vapor at a constant temperature and pressure.

The exercise improvement advice here is to clarify that enthalpy change encompasses both the internal energy change (\(ΔU\)) and the work done by expanding the gas against constant atmospheric pressure (\(PΔV\)). The enthalpy change during the vaporization process provides us with the total amount of energy needed to overcome intermolecular forces within the liquid (related to ΔU) and to push back the atmosphere to make room for the expanding gas (related to work done, \(PΔV\)).

Internal energy of a system, denoted as U, is the sum of the kinetic and potential energies of all particles within the system. It's a microscopic energy on the molecular level. For water, this includes the energy associated with the motion of molecules (kinetic) as well as the energies involved in interactions between molecules (potential).

During vaporization, when water changes from liquid to gas, the internal energy increases due to the greater freedom of movement and distance between water molecules in the gaseous state compared to the liquid state. In the exercise, this increment is represented as the change in internal energy, ΔU. Understanding internal energy is key to grasping why vaporization requires energy: it is needed to break the strong hydrogen bonds between water molecules in the liquid state. The solution shows us how to calculate this internal energy change, highlighting its significant role in the process of vaporization.