Problem 10 - Chemistry An Atoms First Approach

Question

Draw a highly magnified view of a sealed, rigid container filled with a gas. Then draw what it would look like if you cooled the gas significantly but kept the temperature above the boiling point of the substance in the container. Also draw what it would look like if you heated the gas significantly. Finally, draw what each situation would look like if you evacuated enough of the gas to decrease the pressure by a factor of 2

Step-by-Step Solution

Answer

In summary, there are four scenarios to draw to visualize gas behavior in a sealed, rigid container at different temperatures and pressures: 1. Original state: Gas is evenly distributed with temperature (T) and pressure (P). 2. Gas cooled but above boiling point: Gas molecules are closer together with reduced temperature (T_cooled) and slightly reduced pressure (P_cooled). 3. Gas heated significantly: Gas molecules are further apart and appear to move faster with increased temperature (T_heated) and increased pressure (P_heated). 4. Reduced pressure by a factor of 2: For each scenario, draw half the number of gas molecules with respective temperature (T, T_cooled, and T_heated) and half of the original pressure (P/2, P_cooled/2, and P_heated/2).

Step 1: Draw the original state of the gas

To draw the original state, imagine a sealed, rigid container filled with a gas distributed evenly throughout. At this state, the gas molecules are moving at an average speed. Draw a magnified view with circles representing the gas molecules at random positions within the container. Indicate the temperature (T) and pressure (P) to represent the initial conditions.

Step 2: Draw the gas after being cooled but still above boiling point

When the gas is cooled significantly but still above the boiling point, the gas remains in gaseous form and does not condense into liquid. The cooled gas molecules will move more slowly, and they will be closer together. Draw a magnified view of the container with the same number of circles representing the gas molecules, but now they are closer together. Indicate the reduced temperature (T_cooled) and the slightly reduced pressure (P_cooled).

Step 3: Draw the gas after being heated significantly

When the gas is heated significantly, the molecules move more quickly and spread out further apart. Draw a magnified view of the container with the same number of circles representing the gas molecules, but now they are further apart and appear to move faster. Indicate the increased temperature (T_heated) and the increased pressure (P_heated).

Step 4: Draw the situations with reduced pressure by a factor of 2

For each of the three cases described above, draw the same magnified view but now with only half the number of circles representing the gas molecules, since the pressure decreased by a factor of 2. In each case, indicate the respective temperature (T, T_cooled, and T_heated) and the new pressure, which is half of the original pressure in each situation (P/2, P_cooled/2, and P_heated/2). Now, you have successfully drawn all the required scenarios representing the gas behavior at different temperatures and pressures in a sealed, rigid container.

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory of gases provides a framework for understanding the behaviors and properties of gases. It explains how gas molecules are always in motion and how this motion relates to temperature and pressure. According to the theory, gas molecules are constantly moving in random directions. These movements result in collisions with one another and with the walls of their container.

• Molecules in a gas are tiny compared to the distances between them.
• They move in straight lines until they collide with something.
• The temperature of a gas is a measure of the average kinetic energy of its molecules.

This means that when temperature increases, molecules move faster, spreading out more. This concept is crucial when considering how gases behave when heated or cooled within a container.

Temperature and Pressure Relationship

Temperature and pressure are intimately related when it comes to gases, often described by Gay-Lussac's Law. When you increase the temperature of a gas, the molecules gain energy, causing them to move faster. This increased motion means more collisions with the container walls, leading to a rise in pressure.

Conversely, when the temperature is lowered, the molecules slow down, decreasing both the frequency and force of their wall collisions. Consequently, the pressure falls. This relationship can be summarized as:

• Heating a gas → Molecules move faster → Pressure increases
• Cooling a gas → Molecules move slower → Pressure decreases

Understanding this relationship allows us to predict how a gas will behave in different temperature settings, useful for solving problems related to cooling or heating gases in sealed containers.

Phase Changes

Phase changes describe the transformation from one state of matter to another. Although the exercise keeps the gas in the same phase, it's essential to understand how these changes can affect gas behavior.

• When a gas is cooled to its condensation point, it turns into a liquid.
• When a liquid is heated to its boiling point, it turns into a gas.

In our specific exercise, the gas remains above its condensation point due to being properly heated. This is why, despite cooling, there is no phase change. However, understanding at what temperatures these changes might occur is critical for predicting gas behavior in varying conditions.

Gas Laws

Gas laws such as Boyle's Law, Charles's Law, and the Combined Gas Law explain the relationships between pressure, volume, and temperature for a given amount of gas.

By modifying the number of gas particles in the container, as the exercise stipulates, Boyle's Law becomes relevant, which states that pressure is inversely proportional to volume, keeping temperature constant.

• Boyle’s Law: At constant temperature, \[ P_1V_1 = P_2V_2 \]
• Charles's Law: At constant pressure, volume is proportional to temperature.
• Combined Gas Law: Combines Boyle’s and Charles's laws and is given by \[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]

In solving the exercise, one can see that changing the pressure by altering the number of molecules helps illustrate these laws in action, enforcing the conceptual understanding of how gases respond to environmental changes.