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The First Law of Thermodynamics is a foundational principle in physics and chemistry. It states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of an ideal gas, this law can be expressed as: \[ \Delta U = Q - W \] where:

• \( \Delta U \) is the change in the internal energy of the system.
• \( Q \) is the heat added to the system (negative if heat is removed).
• \( W \) is the work done by the system (positive when the system does work on the surroundings).

For the process in the exercise, heat is liberated by the gas, meaning \( Q \) is negative. Therefore, to find the internal energy change, you must subtract the work done by the gas from the negative heat. This concept illustrates the conservation of energy by balancing inflow and outflow within the gas system.

Heat capacity is an important concept that represents how much heat is required to change the temperature of a substance by a certain amount. It depends on whether the gas is at constant volume or constant pressure:

• Constant Volume (\( C_v \)): No expansion occurs, and it reflects the energy required without any work being done by the gas.
• Constant Pressure (\( C_p \)): The gas can expand, and it accounts for both temperature change and work done during expansion.

In this text's scenario, heat is liberated when cooling nitrogen with a constant volume. Therefore, the heat capacity at constant volume \( C_v \) is used. For nitrogen gas, \( C_v \) is approximately 29 J/mol K. It's pivotal in calculating the heat exchange when the volume does not change as it directly relates the temperature change and the heat involved using \( Q = nC_v\Delta T \).

Internal energy \( U \) of a gas is the total energy contained within it. For ideal gases, this energy comes primarily from the kinetic energy of molecules. Internal energy in a gas depends on the number of moles \( n \) and the temperature \( T \), following \[ U = \frac{3}{2}nRT \] where \( R \) is the ideal gas constant.In practical applications, the change in internal energy \( \Delta U \) is often of interest, especially when relating to processes such as cooling or heating like in this exercise, where the gas cools from 50.0°C to 10.0°C. Because temperature drops, there's a loss in internal energy which can be determined using the expression derived from the First Law of Thermodynamics: \[ \Delta U = Q - W \].This change tells us how much less kinetic energy is stored in the gas particles due to the cooling.

The work done by or on a gas is a key concept that helps in understanding energy exchanges during expansion or compression. The work \( W \) done by the gas is calculated for processes at constant pressure using the formula:\[ W = P \Delta V \] where:

• \( P \) is the constant pressure exerted on or by the gas.
• \( \Delta V \) is the change in volume (difference between final and initial volume).

In scenarios where the volume change is related to temperature change, another form of the equation can be used based on the ideal gas law, substituting \( nRT \) for \( PV \). In this exercise, the gas does work as it cools and its volume changes due to temperature decrease. Calculating this work involves understanding the ideal gas relations between temperature, pressure, and volume, combining them to find the extent of energy transferred as mechanical work.