An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings. In such a process, any change in internal energy is solely due to work done on or by the system.
For an ideal gas undergoing an adiabatic process, the relationship between pressure \( P \), volume \( V \), and temperature \( T \) can be described by the following equations:
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Adiabatic Condition (Pressure-Volume Relationship): \[ PV^{\gamma} = \text{constant} \] where \( \gamma = \frac{C_p}{C_v} \) is the heat capacity ratio (specific heat at constant pressure \( C_p \) over specific heat at constant volume \( C_v \)).
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Temperature and Volume Relationship: \[ TV^{\gamma - 1} = \text{constant} \]
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Temperature and Pressure Relationship: \[ TP^{\frac{1 - \gamma}{\gamma}} = \text{constant} \]
In these equations, \( P \) is the pressure, \( V \) is the volume, and \( T \) is the temperature of the gas. Since no heat is transferred in an adiabatic process, the work done on or by the gas changes its internal energy and alters its temperature. An important characteristic of an adiabatic process is that it can result in a temperature change even though no heat flow occurs into or out of the system.
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