A monatomic ideal gas that is initially at 1.52 ×10^5 Pa and has a volume of 7.90 ×10^-2 m^3 i

step 1 So starting off with Part A for question 27, we need to find what is the final pressure? So we h

A monatomic ideal gas that is initially at 1.52 ×10^5 Pa...

A monatomic ideal gas that is initially at $1.52 \times 10^{5} \mathrm{~Pa}$ and has a volume of $7.90 \times 10^{-2} \mathrm{~m}^{3}$ is compressed adiabatically to a volume of $4.50 \times 10^{-2} \mathrm{~m}^{3}$. (a) What is the final pressure? (b) How much work is done by the gas? (c) What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

Key Concepts

Adiabatic Process

An adiabatic process is one in which there is no heat exchange between a system and its surroundings. In thermodynamics, this implies that any change in the internal energy of the system is solely due to work done by or on the system. For an ideal gas undergoing an adiabatic process, the relationship between pressure and volume is given by the equation PV^? = constant.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of an ideal gas. It provides a fundamental basis for understanding gas behavior under various conditions and is often used in conjunction with other thermodynamic equations to analyze processes such as adiabatic compression and expansion.

Specific Heat Ratio (?)

The specific heat ratio, represented by ?, is the ratio of the specific heat at constant pressure (Cp) to that at constant volume (Cv). For a monatomic ideal gas, ? typically equals 5/3. This ratio is crucial in adiabatic processes as it determines the exponent in the adiabatic condition (PV^? = constant) and affects how pressure, volume, and temperature change during the process.

Work Done in Thermodynamic Processes

In thermodynamic processes, work is often calculated by integrating the pressure with respect to volume. For an adiabatic process, because there is no heat transfer, the work done by or on the gas can be directly related to changes in its internal energy and involves integrating the adiabatic pressure-volume relation. The sign of the work determines whether the system is doing work on the surroundings or vice versa.

Temperature Change in Adiabatic Processes

During an adiabatic process, temperature changes occur as a result of the work done on or by the gas. The relation T·V^(??1) = constant (or alternatively, T ? P^((??1)/?)) links the temperature change to variations in volume and pressure. This relationship helps determine whether the gas heats up or cools down during adiabatic compression or expansion.

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