A monatomic gas is adiabatically compressed to ⅛ of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? d. The molar specific heat at constant volume.

Verified step by step guidance

1. Understand the concept of molar specific heat at constant volume (Cv). This is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant volume.
2. Recognize that the molar specific heat at constant volume (Cv) is a property of the material itself and does not depend on the process (such as compression or expansion) the gas undergoes.
3. Identify that in an adiabatic process, no heat is transferred into or out of the system. The change in the system's state (like volume or temperature) does not affect the intrinsic properties of the gas such as Cv.
4. Conclude that since Cv is an intrinsic property and independent of the process, it remains unchanged regardless of the change in volume or temperature of the gas during adiabatic compression.
5. Summarize that the molar specific heat at constant volume (Cv) does not change when a monatomic gas is adiabatically compressed to 1/8 of its initial volume.

Key Concepts

Molar Specific Heat at Constant Volume (Cv)

The molar specific heat at constant volume (Cv) is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the volume constant. For a monatomic ideal gas, Cv is a constant value, specifically 3/2 R, where R is the universal gas constant. This property is intrinsic to the gas and does not change with volume or pressure changes during adiabatic processes.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In such processes, any change in the internal energy of the gas is due solely to work done on or by the gas. For a monatomic gas undergoing adiabatic compression, the temperature increases as the volume decreases, but the specific heat at constant volume remains unchanged since it is a property of the gas itself.

Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. This law helps in understanding how changes in volume and temperature affect the state of the gas. In the context of adiabatic processes, the ideal gas law can be used to derive relationships between these variables, but it does not affect the intrinsic properties like Cv.