91Ó°ÊÓ

The First Law of Thermodynamics is a fundamental principle that describes the conservation of energy. It states that the total energy of a closed system is constant. Mathematically, it can be expressed as \( Q = \Delta U + W \), where:

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

In simple terms, this law explains how energy is transformed but not lost. When heat is added to a gas, it can either increase the gas's internal energy or be used to do work, such as expanding its volume. This transformation of energy aligns with the coal of energy conservation in physics, emphasizing that energy, although capable of changing form, cannot be created or destroyed.

Heat transfer, in the context of thermodynamics, refers to the transfer of thermal energy between a system and its surroundings. In our initial problem, heat \( Q \) flows into a monatomic ideal gas, a perfect illustration of heat transfer. There are three primary modes of heat transfer:

• Conduction,
• Convection,
• Radiation.

Each method involves the movement of thermal energy due to temperature differences. For example, in gases, convection is the most common form, where heated gas molecules expand, rise, and transfer energy. The process is crucial as it contributes to changes in internal energy or results in mechanical work being done by or on the gas, such as expansion.

In physics, a monatomic gas is composed of individual atoms, unlike diatomic or polyatomic gases, which are made up of molecules. Common examples include noble gases like helium and neon. These gases exhibit unique behaviors under heat due to their simple, non-interacting atomic structure. When considering the kinetic theory of gases, the internal energy \( \Delta U \) of a monatomic ideal gas is directly connected to its temperature. For a monatomic ideal gas, the change in internal energy is expressed as \( \Delta U = \frac{3}{2}nR(T_2 - T_1) \), where \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. This relationship is vital in determining how a monatomic gas responds to heat addition, influencing both its internal energy and the work it performs.

Expansion work is the work done by a gas as it expands against an external pressure. In thermodynamics, when a gas expands at constant pressure, the work done \( W \) can be calculated using \( W = P\Delta V \), where \( P \) is the pressure and \( \Delta V \) is the change in volume. In the context of the initial problem, this concept is crucial in calculating how much of the input heat is used for work during the process of gas expansion. The work done by the expansion, using the ideal gas law, can also be represented as \( W = nR(T_2 - T_1) \). Understanding expansion work helps explain how energy from heat can be converted into mechanical energy, showcasing the interaction between thermal energy and physical expansion of gases. This is a fundamental concept in thermodynamics, illustrating the practical use of energy in gases during thermodynamic processes.