Problem 11 - An Introduction to Thermal Physics

Question

Rooms \(A\) and \(B\) are the same size, and are connected by an open door. Room \(A\), however, is warmer (perhaps because its windows face the sun). Which room contains the greater mass of air? Explain carefully.

Step-by-Step Solution

Answer

Room B contains the greater mass of air.

Step 1: Understanding the Problem

Rooms A and B are of the same size and are connected. Room A is warmer due to factors like its windows facing the sun. The task is to determine which room contains the greater mass of air.

Step 2: Applying Ideal Gas Law

The ideal gas law formula, \( PV = nRT \), relates pressure \(P\), volume \(V\), temperature \(T\), and number of moles \(n\) of a gas, where \(R\) is the universal gas constant. At a constant pressure and volume, the number of moles \(n\) is inversely proportional to the temperature \(T\). Since both rooms have the same volume and are connected (same pressure), the room with the lower temperature will have more moles of air, hence greater mass.

Step 3: Comparing Temperatures and Masses

Since Room A is warmer (higher temperature) than Room B, the air in Room A will expand, resulting in fewer moles of air compared to Room B. Consequently, Room B, being cooler, will contain more moles and thus a greater mass of air. This is due to air's density being inversely related to temperature at constant pressure.

Key Concepts

Pressure and Volume

Understanding how pressure and volume relate is crucial when discussing gases. The Ideal Gas Law, expressed as \( PV = nRT \), helps us here. This equation shows the relationship between pressure \( P \), volume \( V \), and the number of moles \( n \) of a gas, with \( R \) being the universal gas constant and \( T \) the temperature. At constant temperature, increasing the volume of a container will decrease the pressure if the number of gas moles remains constant. Conversely, decreasing the volume will increase the pressure. This inverse relation is known as Boyle's Law, a component of the Ideal Gas Law. When rooms are connected, as in the exercise, they tend to equalize in pressure if volume and temperature differences aren’t significantly altered. Since rooms \( A \) and \( B \) share a passage, they have an equal pressure if no other outside forces affect them. This condition is why the comparison focuses on temperature differences in this scenario.

Temperature and Density

Temperature influences the density of gases in an interesting way. Density is defined as mass per unit volume. In the context of gases and the Ideal Gas Law, if temperature increases, the gas molecules move more rapidly, causing them to occupy more space, and effectively decreasing density. This is why Room \( A \), being warmer, has less air density than Room \( B \). Since density decreases with increasing temperature at a constant pressure (the condition in our connected rooms), less dense air means fewer molecules are present in Room \( A \). Therefore, Room \( B \), being cooler, has air that is denser and more massed. This concept is central to understanding why the mass of air differs between two connected rooms at different temperatures.

Moles of Air

The concept of moles is crucial to understanding how much gas is present in a given space. In the Ideal Gas Law, \( n \), the number of moles, indicates the quantity of gas present. At a constant volume and pressure, temperature largely determines successful application of the Ideal Gas Law when comparing two spaces or containers. For Room \( A \) and Room \( B \), the room with a higher temperature, Room \( A \), contains fewer moles of air. Since the number of moles is inversely proportional to temperature, Room \( B \), being cooler, contains more moles of air. More moles translate to more mass, hence why Room \( B \) effectively has a greater air mass. These relations help explain various real-world phenomena, such as balloon deflation in cooler temperatures or how air conditioning can affect air density.