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 thermal energy of the gas.

Verified step by step guidance

1. Understand the context: In an adiabatic process, no heat is exchanged with the surroundings (Q = 0). For a monatomic ideal gas, the internal energy (thermal energy) is directly related to its temperature, and any change in thermal energy comes from work done on or by the gas.
2. Recall the first law of thermodynamics: ΔU = Q - W. Since Q = 0 in an adiabatic process, the change in internal energy (ΔU) is equal to the work done on the gas (W). This means the thermal energy of the gas changes as work is done during compression.
3. Use the adiabatic condition: For an adiabatic process, the relationship between pressure, volume, and temperature is governed by the equation P V^γ = constant, where γ (gamma) is the adiabatic index. For a monatomic gas, γ = 5/3.
4. Relate temperature to volume: The temperature of the gas changes according to the equation T₂/T₁ = (V₁/V₂)^(γ-1), where T₁ and T₂ are the initial and final temperatures, and V₁ and V₂ are the initial and final volumes. Substitute V₂ = V₁/8 and γ = 5/3 to find the factor by which the temperature changes.
5. Connect thermal energy to temperature: The thermal energy of a monatomic ideal gas is given by U = (3/2) nRT, where n is the number of moles and T is the temperature. Since U is proportional to T, the change in thermal energy is directly proportional to the change in temperature. Use the factor derived in the previous step to determine how much the thermal energy increases or decreases.

Key Concepts

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In the context of a gas, this means that any change in the internal energy of the gas is due solely to work done on or by the gas. During adiabatic compression, the gas's volume decreases, leading to an increase in pressure and temperature, while the thermal energy changes as a result of work input.

Thermal Energy

Thermal energy refers to the total kinetic energy of the particles in a substance, which is directly related to its temperature. For an ideal monatomic gas, thermal energy can be expressed as a function of temperature and the number of particles. In an adiabatic process, when the gas is compressed, its thermal energy increases due to the work done on it, resulting in a rise in temperature.

First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In an adiabatic process, since no heat is exchanged, the change in internal energy is equal to the work done on the gas. This principle helps to understand how the thermal energy of the gas changes during compression, as it directly correlates with the work input.