91Ó°ÊÓ

The First Law of Thermodynamics is a fundamental principle in physics that describes the conservation of energy within a closed system. In simpler terms, it tells us how energy enters or leaves a system as a form of heat or work. The law is expressed mathematically as \( \Delta U = Q - W \), where:

• \( \Delta U \) represents the change in internal energy of the system.
• \( Q \) is the heat added to the system.
• \( W \) is the work done by the system.

For instance, if a gas receives heat \( Q \), and it performs work \( W \) by expanding, the change in its internal energy is the heat received minus the work done. This principle helps us understand how energy is transformed from one type to another, such as converting heat into work. In the context of the problem, 90 J of heat is added, and the gas expands at constant pressure, causing changes in its internal energy.

Diatomic gases consist of molecules made up of two atoms, such as oxygen \((O_2)\) and nitrogen \((N_2)\). These gases possess unique properties due to their molecular structure. In thermodynamics, understanding the behavior of diatomic gases is essential because of their additional degrees of freedom compared to monatomic gases.A diatomic gas can not only translate (move) in three directions like a monatomic gas, but it can also rotate. This extra rotational capability affects how energy is distributed in the system. For diatomic gases, the degree of freedom plays a significant role in determining how its internal energy will change with heat input and work done during processes such as expansion or compression. Hence, when heat is added to a diatomic gas, as mentioned in our problem, it directly impacts its internal energy differently compared to monatomic gases.

Internal energy is the total energy contained within a system due to the random motion and potential energies of its particles. In a gas, this is primarily due to the kinetic energy of the molecules moving around. When a gas receives heat or performs work, its internal energy changes. For a diatomic gas, which can both translate and rotate, its internal energy increase is affected by the heat and the degrees of freedom available. Consequently, when a diatomic gas such as the one in our exercise is heated by 90 J, the increase in internal energy is determined by its degrees of freedom, affecting how that energy is partitioned among translational and rotational modes without involving vibrational modes, as those are inactive in this problem.

Degrees of freedom in thermodynamics refer to the independent modes by which a system's molecules can store energy. For a diatomic gas, these include:

• Three translational motions (movement in three-dimensional space).
• Two rotational motions (rotation around two perpendicular axes).

Thus, a diatomic gas in normal conditions has five degrees of freedom. However, vibrational motion, which would add another degree, is not active at lower temperatures or the conditions specified in some problems. These degrees allow the calculation of a gas's internal energy change when heat is added. Using the formula \( \Delta U = \frac{f}{f+2} Q \), where \( f \) is the degree of freedom, tells us that only a fraction of the heat contributes to the internal energy, as seen in our exercise where \( \frac{5}{7} \) of the 90 J of heat increases the internal energy. Understanding degrees of freedom helps explain not just how much internal energy changes, but also the behavior of gases under various thermodynamic processes.