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Internal energy is a core concept of thermodynamics that represents the total energy contained within a system. In an ideal gas, this energy is related to the random motion of the gas molecules. In our problem, the gas is diatomic, meaning it consists of two atoms per molecule, like oxygen or nitrogen. This structure provides it with five active degrees of freedom at the given temperatures. These include three translational movements and two rotational movements.

To determine the change in internal energy (\( \Delta U \)), we can use the equation:

• \( \Delta U = \frac{5}{2} nR \Delta T \)

Here, \( n \) denotes the number of moles, \( R \) is the ideal gas constant, and \( \Delta T \) is the change in temperature from the initial to the final state. This formula stems directly from the kinetic theory of gases, considering the heat capacity at constant volume for diatomic gases. As the temperature chance dictates the change in internal energy, no work (pressure-volume work) directly affects \( \Delta U \). Thus, the internal energy change is solely linked to the temperature difference of the system.

When considering work done by or on a gas, it's important to understand that this is related to the expansion or compression of the gas. The work (\( W \)) done by the gas is formulated as:

• \( W = \int P \, dV \)

This expression calculates the area under the pressure-volume (\( P-V \)) graph for the specific process. Plotting pressure versus volume typically results in a curve or line, depending on how pressure relates to temperature and volume.

In the given exercise, pressure increases linearly with temperature. This relationship can usually be represented as a linear equation, \( P = mT + b \), where \( m \) is the slope obtained from the change in pressure and temperature, and \( b \) is the intercept. By integrating this linear equation over the changes in volume, we can calculate the total work done by the gas during its transition from the initial to the final state. This mathematical evaluation takes into consideration the path taken by the process within the \( P-V \) plane.

Heat transfer in a thermodynamic process describes how energy is exchanged between a system and its surroundings due to temperature differences. In the context of ideal gases, heat transfer (\( Q \)) is linked with both the change in internal energy and the work done by the system.

The first law of thermodynamics establishes the relation:

• \( \Delta U = Q - W \)

This equation states 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. In practical terms, if we know \( \Delta U \) from the change in the state of the system and \( W \) from work calculations, we can easily solve for the heat added using this fundamental relationship. Understanding this concept is crucial in identifying how systems exchange energy and how processes such as heating or cooling impact a gas at molecular levels.

Degrees of freedom in the context of ideal gases refer to the number of independent ways in which the gas molecules can move. For a diatomic gas, this involves translational and rotational motion. These degrees are crucial in determining how energy is partitioned within a gas.

Let's break it down:

• Translational movement: 3 degrees (movement along x, y, z axes)
• Rotational movement for diatomic molecules: 2 degrees (rotation around two perpendicular axes)

Adding these up, the gas has 5 degrees of freedom. This is important because the internal energy of a gas depends on these degrees; specifically, each degree of freedom contributes equal amounts of energy, amounting to \( \frac{1}{2}k_B T \) per molecule per degree of freedom (where \( k_B \) is Boltzmann's constant). The enumeration of degrees of freedom is pivotal in correctly applying formulas related to energy calculations, such as determining the internal energy change in a gas.