Question

For the process \(\mathrm{A}(l) \longrightarrow \mathrm{A}(g),\) which direction is favored by changes in energy probability? Positional probability? Explain your answers. If you wanted to favor the process as written, would you raise or lower the temperature of the system? Explain.

Short answer

The given process A(l) → A(g) is favored by both energy probability and positional probability as it requires sufficient energy to break the intermolecular forces, and transitioning from liquid to gas increases the positional probability. To favor this process, the temperature of the system should be raised, as it will provide the necessary energy to increase the energy probability for the forward reaction.

Step-by-step solution

Understand the process

We are given a process in which substance A changes its state from liquid (l) to gas (g). We need to determine whether this process is favored by energy probability and positional probability.

Analyze energy probability

Energy probability refers to the likelihood of a system adopting a certain energy state. For a system to transition from one state to another, it needs to overcome the potential energy barrier between those states. In the case of a liquid-to-gas transition, the system needs to overcome the intermolecular forces that hold the molecules together in the liquid state. When the energy probability favors the forward reaction (A(l) → A(g)), it means that the system has enough thermal energy to break the intermolecular forces of the liquid state. As a result, the molecules gain freedom to move as a gas. In this case, the forward reaction (liquid to gas) is favored by energy probability. Therefore, energy probability favors the direction from liquid to gas.

Analyze positional probability

Positional probability refers to the likelihood of a particle being found in a certain location or state. For a substance to transition from a more ordered to a less ordered state, the positional probability favors the process. When going from liquid to gas, the substance becomes less ordered and the particles have more freedom to move. As a result, the positional probability favors the forward reaction (A(l) → A(g)) as it allows the particles to occupy more positions in space, increasing the entropy of the system. Therefore, positional probability favors the direction from liquid to gas.

Determine the effect of temperature on the process

To favor the given process (A(l) → A(g)), we need to ensure that the energy probability and positional probability are favored. As we concluded in steps 2 and 3, both energy and positional probabilities favor the forward reaction. However, we need to know how temperature affects this process. As the forward reaction (A(l) → A(g)) requires energy to break the intermolecular forces, increasing the temperature provides this energy and increases the energy probability for the forward reaction. Therefore, we should raise the temperature of the system to favor the process as written.

Key concepts

Energy Probability

Energy probability looks at how likely a system is to adopt certain energy states. When we talk about the liquid-to-gas transition, like in the process \( \mathrm{A}(l) \rightarrow \mathrm{A}(g) \), we need to conquer the intermolecular forces holding the liquid together. These forces are like tiny hooks keeping the molecules from drifting apart into a gas.
For the system to move from a liquid to a gas, it needs enough thermal energy to break those hooks or forces.
This is where energy probability comes into play. If the system has enough energy, it makes it easier for molecules to escape into the gaseous state, meaning the forward reaction is favored.

• Energy probability favors states with sufficient energy.
• The forward reaction (liquid to gas) is favored when thermal energy overcomes intermolecular forces.

In essence, the energy probability tells us that with enough energy, molecules will prefer spreading out into a gas, making this direction favorable.

Positional Probability

Positional probability examines how particles spread out in space. Imagine particles in a liquid as people sitting in tightly packed rows. When it turns into a gas, it’s like those people spreading out into an open field, free to move around.
This transition means the system becomes less ordered and the particles can occupy more positions, increasing entropy. In simpler terms, as particles move from being orderly packed in a liquid to freely dispersed in a gas, they enjoy more freedom, akin to moving from a confined room to an open garden.

• Positional probability increases when particles occupy more space.
• The transition from liquid to gas is favored because it increases positional probability.

Thus, positional probability naturally supports the forward reaction from liquid to gas because it leads to greater randomness and disorder.

Temperature Influence

Temperature plays a crucial role in enhancing the chances of a phase transition. When deciding if we should raise or lower the temperature for the process \( \mathrm{A}(l) \rightarrow \mathrm{A}(g) \), temperature becomes a decisive factor in boosting energy probability.
Higher temperatures infuse additional energy into the system. This energy helps the molecules overcome the forces that bind them in the liquid state. Effectively, raising the temperature quickens the process of breaking these forces, making it easier for the molecules to disperse as gas.

• Raising temperature increases thermal energy.
• Increased energy assists in breaking liquid's intermolecular forces, favoring gas formation.

Therefore, by raising the temperature, both energy and positional probabilities are effectively maximized, ensuring the process is favored.