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The internal energy of a system is a measure of the total energy contained within it. For an ideal gas, this energy comprises solely of the kinetic energy of the particles that make up the gas.

When considering the internal energy change (abla U) of a gas, it is crucial to remember that it relates to heat (Q) and work (W) done by or on the system. The formula to express this change in an ideal gas is given by:

• \(\Delta U = Q - W\)

This equation states that the change in internal energy is equal to the heat added to the system minus the work done by the system. However, if work is done on the system, as in the exercise, we consider W as negative since it is added energy.

In the given problem, calculating the change in internal energy is a crucial step, where we add both the heat introduced and the work done to find the overall change.

When we talk about a monatomic gas, we refer to a gas composed of single-atom molecules. Examples of monatomic gases include noble gases like helium and neon. Monatomic gases are often idealized because of their simple nature, making calculations involving them straightforward.

The behavior of monatomic gases can be explained using the Ideal Gas Law and principles of thermodynamics. For such gases, the degrees of freedom come into play—monatomic gases have only translational degrees of freedom, three in total.

When applying the internal energy formula to a monatomic gas:

• \( \Delta U = \frac{3}{2} n R \Delta T \)

we consider these three degrees of freedom. The variable n represents the number of moles, and R is the ideal gas constant. This equation helps calculate how temperature and energy interchange affect the gas.

Temperature change (\Delta T) is an essential aspect of thermodynamic problems involving gases. It gives insight into how the energy input into a gas impacts its thermal state.

When heat is added to a gas or work is performed on or by the gas, the temperature of the gas changes, provided there is no phase change. For a monatomic ideal gas, the connection between temperature change and internal energy is particularly straightforward:

• \( \Delta T = \frac{2 \Delta U}{3 n R} \)

This equation allows us to isolate the temperature difference resulting from any given change in internal energy. By substituting known values, one can compute how much the temperature of the gas has increased or decreased.

In the problem, after determining the change in internal energy, this formula helps convert that energy change into a temperature change.

Heat and work are two critical concepts in thermodynamics as they describe energy transfer between systems and their environment. However, they are not forms of energy; instead, they are processes of energy transfer.

**Heat ( Q ):** Heat is energy transferred due to temperature differences. In the context of an ideal gas, when heat is delivered to it, the energy is transferred to the gas particles, generally increasing their kinetic energy, thereby raising the internal energy and perhaps increasing the temperature.

• Positive Q means energy is added to the gas.

**Work ( W ):** Work involves energy being transferred through forces acting across a distance. For gases, this often involves the gas expanding or contracting. For this exercise, work is done **on** the gas, meaning energy is added to the system, which is why W is considered negative.

• Negative W implies work is done on the gas, increasing its internal energy.

Understanding these forms of energy transfer allows us to manipulate and solve the equations effectively, as shown in calculating the final temperature.