Question

Which process is spontaneous at 298 \(\mathrm{K}\) ? a. \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g, 1 \mathrm{atm})\) b. \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g, 0.10 \mathrm{atm})\) c. \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g, 0.010 \mathrm{atm})\)

Short answer

The spontaneous process at 298 K is option c. \(\mathrm{H}_{2}\mathrm{O}(l) \longrightarrow \mathrm{H}_{2}\mathrm{O}(g, 0.010 \mathrm{atm})\)

Step-by-step solution

Recall the definition of spontaneity

A spontaneous process is one that occurs without needing to be driven by an external force or energy. In thermodynamics, a process is considered spontaneous if it leads to an increase in the entropy of the universe.

Understand the relationship between vapor pressure and spontaneity

For the conversion of liquid water to gaseous water (vapor), the process is spontaneous if the vapor pressure of the water is greater than the external pressure. This is because spontaneous evaporation occurs when the molecules in the liquid phase have enough kinetic energy to overcome the atmospheric pressure.

Analyze the given options for their vapor pressures

At 298 K, the standard vapor pressure of water is about 0.03 atm. Therefore, any situation where the external pressure is less than this vapor pressure would result in a spontaneous process.

Determine the spontaneous process

Option (a) has an external pressure equal to 1 atm, which is greater than the vapor pressure of water at 298 K, thus the process is not spontaneous. Option (b) has an external pressure of 0.10 atm, which is greater than the vapor pressure of water at 298 K, so this process is also not spontaneous. Option (c) has an external pressure of 0.010 atm, which is less than the vapor pressure, so this process would be spontaneous.

Key concepts

Spontaneity Definition

Understanding the concept of spontaneity is fundamental in thermodynamics and guides us in predicting the behavior of processes without the need for external intervention.

At its core, a spontaneous process is one that can proceed on its own, given initial conditions, without persistent input from external sources. For a process to be regarded as spontaneous, it does not necessarily have to be fast or occur immediately; what matters is the inherent potential for the process to proceed autonomously at a given temperature and pressure.

It's imperative to grasp that spontaneity in thermodynamics is deeply connected to the concept of entropy. According to the Second Law of Thermodynamics, the total entropy of the universe must increase for a process to be spontaneous. That is, the overall disorder at a universal scale, which includes both the system and its surroundings, must rise.

This fundamental principle helps us discern the direction of natural processes and their feasibility, giving rise to predictions about what reactions or phase changes can occur without external energy inputs.

Vapor Pressure and Spontaneity

The conversion of liquids to gases offers a compelling illustration of spontaneous processes, particularly considering the role of vapor pressure. Vapor pressure is characterized by the tendency of molecules within a liquid to escape into the gas phase at a specific temperature.

This pressure is a measure of the escaping tendency of the liquid molecules and is influenced by both temperature and the identity of the liquid. A crucial factor in determining spontaneity for a phase transition from liquid to gas is comparing the liquid's vapor pressure to the external or atmospheric pressure.

When the vapor pressure surpasses atmospheric pressure, the liquid molecules have sufficient kinetic energy to escape into the gas phase — a spontaneous process. Conversely, if the atmospheric pressure exceeds the vapor pressure, additional energy is required to promote phase change, thus it's non-spontaneous. Therefore, in the given exercise, figuring out which instance led to spontaneous vaporization of water hinged on this delicate balance between vapor pressure and external pressure.

Entropy and Spontaneity

The concept of entropy is synonymous with the degree of randomness or disorder within a system. When we speak of spontaneous processes, we often correlate them with an increase in entropy.

For a process to be considered spontaneous, there must typically be an overall increase in entropy, which includes both the system and its surroundings. This doesn't always mean that the entropy of the system alone must increase; the surrounding's entropy change can compensate for the system's entropy change, provided the total entropy change for the universe is positive.

This relationship is beautifully reflected when considering phase transitions. The transition from liquid to gas increases the entropy of the system because the gas particles are more disordered than those in the liquid state. For spontaneous evaporation, as in our textbook exercise, the lowering of external pressure allows the entropy of the liquid to increase, ushering in a gas phase as a spontaneous phenomenon.

These fundamental principles not only underpin the vast field of thermodynamics but also give us a predictive edge regarding the behavior of materials and reactions in various conditions, weaving a coherent story of spontaneity tied to entropy changes and vapor pressures.