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The First Law of Thermodynamics is a pivotal principle in understanding energy changes within a system. It states that the total energy in an isolated system is constant, implying energy cannot be created or destroyed but can change forms. This principle can be mathematically represented as: \[ \Delta U = Q - W \] where

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

In our problem, the experimenter added 970 J of heat (\( Q \)) to the system, and the gas performed 223 J of work (\( W \)). Thus, the change in internal energy \( \Delta U \) was calculated as 747 J. This is a straightforward application of the First Law, illustrating how energy transfers and transformations balance within the gas.

An ideal gas is a theoretical gas that perfectly follows the ideal gas laws without any deviations. It's an important concept because it provides a simplified but very useful model of how gases behave under different conditions. The characteristics of an ideal gas include:

• Particles that have negligible size
• No intermolecular forces between particles, meaning interactions only occur during collisions
• Elastic collisions, meaning kinetic energy is conserved

Ideal gases are defined by the equation of state known as the Ideal Gas Law:\[ PV = nRT \]where

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

In the given problem, the ideal gas assumption allows us to directly relate the heat added and work done to the change in internal energy, using simplified relations valid for ideal gases.

Internal energy is the total energy contained within a system, originating from the microscopic movements and interactions of its molecules. For an ideal gas, this internal energy primarily depends on temperature because:- It encompasses kinetic energy of particles, which is directly related to temperature.- The formula for change in internal energy of an ideal gas at constant volume can be simplified to: \[ \Delta U = nC_v \Delta T \]In this scenario, the gas's internal energy increased by 747 J when heated from 10.0°C to 25.0°C. This change was determined using the First Law of Thermodynamics, confirming that the internal energy change was due to both the heat added and the work performed by the gas during expansion.

Specific heat capacity, a critical concept in thermodynamics, refers to the amount of heat required to change the temperature of a substance per unit mass. When dealing with gases, we often use molar heat capacities:

• \( C_p \): specific heat capacity at constant pressure
• \( C_v \): specific heat capacity at constant volume

For our ideal gas, these were calculated as follows:- \( C_p = \frac{Q}{n\Delta T} \approx 37.14 \text{ J/mol·K} \)- \( C_v = \frac{\Delta U}{n\Delta T} \approx 28.46 \text{ J/mol·K} \)The ratio of these two, known as the adiabatic index (\( \gamma \)), was found to be 1.30. This index is significant because it provides insights into how gases expand and compress without heat exchange, influencing the speed of sound in the gas and other dynamic properties.