A monatomic gas is adiabatically compressed to 1/8 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? The molar specific heat at constant volume.

Verified step by step guidance

1. Step 1: Begin by understanding the concept of molar specific heat at constant volume (Cv). For a monatomic gas, Cv is a constant value determined by the degrees of freedom of the gas molecules. Specifically, Cv = (3/2)R, where R is the universal gas constant.
2. Step 2: Recognize that Cv is a property of the gas itself and depends only on the type of gas and its molecular structure. It does not depend on external conditions such as volume, pressure, or temperature.
3. Step 3: Recall that in an adiabatic process, there is no heat exchange with the surroundings (Q = 0). While other thermodynamic quantities like pressure, temperature, and volume may change during adiabatic compression, Cv remains unchanged because it is intrinsic to the gas.
4. Step 4: Conclude that the molar specific heat at constant volume (Cv) does not change during the adiabatic compression of the monatomic gas. This is because Cv is a fixed property of the gas and is independent of the process being performed.
5. Step 5: Summarize that the molar specific heat at constant volume remains constant and does not increase or decrease, nor does it change by any factor during the adiabatic compression.

Key Concepts

Molar Specific Heat at Constant Volume (C_v)

The molar specific heat at constant volume (C_v) 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, C_v 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 under adiabatic processes.

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. In such processes, any change in internal energy is solely due to work done on or by the system. For a monatomic gas undergoing adiabatic compression, the temperature increases as the volume decreases, but the specific heat capacities remain constant, as they are properties 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 specific heat.