An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume \(V_{1}\) and contains an ideal gas at pressure \(P_{1}\) and temperature \(T_{1}\). The other chamber has volume \(V_{2}\) and contains the same ideal gas at pressure \(P_{2}\) and temperature \(T_{2}\). If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be (a) \(\frac{T_{1} T_{2}\left(P_{1} V_{1}+P_{2} V_{2}\right)}{P_{1} V_{1} T_{2}+P_{2} V_{2} T_{1}}\) (b) \(\frac{P_{1} V_{1} T_{1}+P_{2} V_{2} T_{2}}{P_{1} V_{1}+P_{2} V_{2}}\) (c) \(\frac{P_{1} V_{1} T_{2}+P_{2} V_{2} T_{1}}{P_{1} V_{1}+P_{2} V_{2}}\) (d) \(\frac{T_{1} T_{2}\left(P_{1} V_{1}+P_{2} V_{2}\right)}{P_{1} V_{1} T_{1}+P_{2} V_{2} T_{2}}\)

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and moles (n) of an ideal gas. The equation is given by \( PV = nRT \), where R is the universal gas constant.

Understanding this law is crucial when predicting how gases will behave under different conditions. For example, increasing the temperature while keeping the volume constant will cause an increase in pressure. Conversely, if the temperature is held constant, increasing the volume will decrease the pressure.

Application to Equilibrium

When applied to our exercise, the ideal gas law is used to determine the relationship between the initial states of the gas in each chamber and the final equilibrium state. This allows us to express the energy in terms of pressure and volume, facilitating the calculation of the final temperature after removing the partition.

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In the context of an isolated system like our gas in two chambers, this means the total energy before and after the removal of the partition must remain the same.

This concept is key to solving for the final equilibrium temperature of the gas in the container, as we assume that no energy has entered or left the system. Therefore, the sum of the energies in both chambers initially should equal the total energy of the combined volume at the final temperature.

Energy Conservation in the Ideal Gas

For ideal gases, since no work is done and no heat is exchanged in the process described in the exercise, the conservation of energy directly relates the initial temperatures and volumes to the final equilibrium temperature.

Internal energy refers to the total energy contained within a system due to the motion and interaction of its molecules. For an ideal gas, the internal energy is primarily kinetic and depends on the temperature of the gas.

One of the key insights from thermodynamics is that for an ideal gas, the internal energy change is proportional to the change in temperature. Lesson content might include the formula \( E = C V T \) for the internal energy \( E \), where \( C \) is the heat capacity, \( V \) is the volume, and \( T \) is the temperature of the gas.

Connecting Temperature and Internal Energy

When dealing with our exercise, understanding the relationship between internal energy and temperature allows us to express the total initial energy and the final energy in terms of the respective temperatures, which is crucial for finding the final equilibrium temperature after the partition is removed.

Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy. The laws of thermodynamics govern the principles of energy conversion and provide frameworks for understanding phenomena like the behavior of an ideal gas.

The Four Laws of Thermodynamics

In this exercise, we can consider the First Law of Thermodynamics, which is another way to state the conservation of energy. It asserts that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system.

• Zeroth Law: Establishes thermal equilibrium and temperature.
• First Law: Describes the conservation of energy.
• Second Law: Indicates the direction of natural processes and defines entropy.
• Third Law: Relates to the impossibility of reaching absolute zero.

In the given problem, we essentially apply the First Law to predict the outcome when the condition of thermal equilibrium is achieved between the two chambers, which tells us about the final temperature once the gas settles into a new equilibrium state.