On an atomic basis, and given that the average kinetic energy of motion of molecules in the gas phase is given by \(\mathrm{KE}=3 R T / 2\), explain why the pressure of a gas doubles when its temperature in Kelvin is doubled. (Hint: Pressure is due to collisions of molecules on the container walls and the force of those collisions.)

Understanding the relationship between pressure, volume, temperature, and the number of gas molecules involves the ideal gas law, which is a cornerstone of chemical and physical gas theories. The law is often expressed as the equation \( PV = nRT \), where \( P \) represents the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

When dealing with problems like the doubling of gas pressure when the temperature is doubled, the ideal gas law provides insight into what happens to a gas under constant volume and a fixed amount of gas. By holding the volume and number of moles constant and focusing on the pressure and temperature, we simplify the equation to \( P \propto T \)—directly highlighting the temperature-pressure relationship at play.

The ideal gas law is a theoretical construct that assumes gases are composed of many tiny particles that are far apart from each other relative to their size. This law also assumes that gas particles are in constant random motion and collide with each other and the walls of the container in a perfectly elastic way, meaning there's no energy loss in collisions. These ideal conditions help us easily relate physical properties of gases, although in reality, gases may not always behave ideally, especially under high pressures or at very low temperatures.

Molecular kinetic energy is the energy that gas molecules possess due to their motion. It's vital for understanding gas behavior, especially when looking at the causes of pressure. The average kinetic energy \( (KE) \) of a molecule in the gas phase is given by the formula \( KE = \frac{3}{2}RT \), where \( R \) is the gas constant and \( T \) is the temperature in Kelvin.

The Role of Kinetic Energy in Pressure

When gas molecules move and collide with the walls of their container, they exert a force, and this repeated action results in pressure. The kinetic theory of gases explains that the pressure exerted by a gas held at a constant volume increases with the temperature because the molecules move faster and therefore, hit the walls more forcefully and more often. It's this molecular movement and collision that underscores the direct proportionality of kinetic energy to pressure.

In our textbook problem, understanding that the kinetic energy of molecules dictates their pressure on a container's walls makes the concept of temperature affecting pressure much clearer. Doubling the kinetic energy (by doubling the temperature) inevitably leads to doubling the pressure, as both are proportionally related.

The temperature-pressure relationship is an important aspect of gas laws that describes how changes in the temperature of a gas, assuming a constant volume and amount of gas, lead to changes in pressure. As we saw in the textbook exercise, when the temperature of a gas is doubled (while keeping the volume constant), the pressure also doubles. This phenomenon is directly linked to the kinetic theory of gases which states that an increase in temperature will result in an increase in the average kinetic energy of the gas molecules.

Explaining the Proportional Increase

If the kinetic energy increases, the molecules move faster and strike the walls of the container more frequently and with greater force, translating into higher pressure. It's this foundational concept that explains why heating a can of air (or any gas) can lead to an increase in pressure that might even cause the can to explode if the temperature rises sufficiently.

Moreover, this relationship helps us predict and calculate changes in gas behavior in various conditions. It's crucial for students to grasp this concept as it explains many natural phenomena and is fundamental to fields such as meteorology, engineering, and physics. The direct relationship makes it simple to predict that if the temperature of a gas is halved, the pressure it exerts (provided volume and moles of gas remain unchanged) will also be halved.