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The change in internal energy, denoted as \( \Delta U \), reflects how much energy is absorbed or released within a system during a process, like the evaporation of water. During a phase change, such as when water transforms from liquid to vapor, energy is needed to overcome intermolecular forces. This energy input does not contribute to a temperature change but rather alters the state of the substance.
In the exercise, we calculate \( \Delta U \) using the formula:

• \( \Delta U = \Delta H - P\Delta V \)

Here, \( \Delta H \) is the enthalpy change during vaporization, and \( P\Delta V \) accounts for the work done by the system as it expands, governed by the ideal gas approximation. Understanding internal energy change is key in thermodynamics, as it provides insight into the energy balance and behavior of substances under different conditions.

Enthalpy of vaporization, \( \Delta H_{\text{vap}} \), is the energy required to convert a substance from a liquid to a gas at constant temperature and pressure. For water, this process occurs at 100°C or 373K, during which each mole of water absorbs heat.

• Given that \( \Delta H_{\text{vap}} = 41 \text{ kJ/mol} \) for water, this value is intrinsic to water's ability to vaporize.

Enthalpy changes, like \( \Delta H_{\text{vap}} \), are significant because they reflect the strength of intermolecular forces; in water's case, strong hydrogen bonds. Evaluating \( \Delta H_{\text{vap}} \) allows us to quantify the energy needed for phase changes in chemical processes and infer the energy dynamics involved.

The ideal gas approximation simplifies calculations by assuming certain conditions under which a gas behaves perfectly. For gases, this means they're considered to have:

• Elastic collisions
• No intermolecular forces
• Infinitesimal particle volume

For the problem at hand, the ideal gas law, \( PV = nRT \), aids in calculating work done by the system as water vaporizes, expressed in the term \( P\Delta V = nRT \). Given that \( \Delta T = 0 \) (as the temperature remains constant during vaporization), the term simplifies calculations without loss of accuracy.While real gases exhibit behaviors deviating from this model, at high temperatures and low pressures, the ideal gas approximation can effectively predict gas characteristics. This framework is integral to problem-solving in thermodynamics where approximations are necessary for practical calculations.