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The Ideal Gas Law is a fundamental principle in thermodynamics that describes the behavior of gases under various conditions. It is given by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin

In this exercise, we are dealing with one mole of a gas (\( n=1 \)), simplifying our equation to \( PV = RT \).
This law helps predict how changes in one of these properties affect the others. For example, when we heat neon gas while keeping the volume steady, the pressure rises, showing how tightly linked these properties are.
Understanding this relationship is crucial for predicting gas behavior in different thermodynamic processes.

Monatomic gases are composed of single-atom molecules like helium or neon. These gases often behave like ideal gases, especially at standard conditions. One special characteristic of monatomic gases is their heat capacity, which distinguishes them from diatomic or polyatomic gases. When considering their heat capacity values, monatomic gases are simple systems. Their specific heat capacity at a constant volume is \( C_v = \frac{3}{2}R \), where \( R \) is the ideal gas constant.
This characteristic is particularly important in thermodynamic calculations, such as determining the energy absorbed or released during heat transfer.

Heat Capacity is a crucial property in thermodynamics, representing the amount of heat needed to change a substance's temperature by a certain amount. For gases, it varies depending on how they are held during heating, either at constant volume or constant pressure.

• Constant Volume Heat Capacity \( (C_v) \): In monatomic gases, \( C_v \) is typically \( \frac{3}{2}R \). This value is used when the gas is heated without any change in volume.
• Constant Pressure Heat Capacity \( (C_p) \): For monatomic gases, \( C_p \) equals \( \frac{5}{2}R \). This applies when the gas is allowed to expand, like when volume changes at a constant pressure.

The distinction between \( C_v \) and \( C_p \) forms the basis for many calculations, such as those determining the total heat transferred to or from the gas.

In a Constant Volume Process, the volume stays the same while other properties like pressure and temperature can vary. For an ideal gas like neon, this situation is common when the gas is heated inside a sealed, non-expandable container.
Since no work is done on or by the gas (work is done by volume changes), the energy transferred as heat results in temperature and pressure changes. In our specific scenario, as the neon is heated, its pressure triples, and the temperature rises accordingly from \( T_1 \) to \( T_2 \).
The heat added, \( Q \), can be calculated using the formula \( Q = nC_v\Delta T \), reinforcing the significance of heat capacity in thermal processes.

During a Constant Pressure Process, the gas pressure remains fixed while the volume and temperature can change. This is typical when a gas is allowed to expand or is compressed in an environment where pressure doesn't change, such as in an open container or system connected to a constant pressure reservoir.
In our exercise, after the initial constant volume heating, neon experiences an expansion, doubling its volume. During this process, the temperature elevates from \( T_2 \) to \( T_3 \), and the heat transferred is calculated using \( Q = nC_p\Delta T \).
This process illustrates how heat capacity at constant pressure (\( C_p \)) is critical in assessing the energy required for temperature changes under constant pressure conditions.