Question

For the adiabatic expansion of an ideal gas: (a) \(\mathrm{PV}^{\gamma}=\) constant (b) \(\mathrm{TV}^{\gamma-1}=\) constant (c) \(\mathrm{T}^{\top} \mathrm{P}^{1-\gamma}=\) constant (d) All of these

Short answer

All of these are correct for an adiabatic expansion of an ideal gas.

Step-by-step solution

Understanding the Terms

Adiabatic expansion involves a process where no heat is exchanged with the surroundings. In these processes for an ideal gas, specific relationships between Pressure (P), Volume (V), and Temperature (T) are maintained. The constant \(\gamma\) is the heat capacity ratio \(C_p/C_v\).

Evaluating Option (a)

In an adiabatic process, the formula \(PV^\gamma = \ ext{constant}\) is a fundamental equation known as the adiabatic condition for pressure and volume.

Evaluating Option (b)

The equation \(TV^{\gamma-1} = \ ext{constant}\) is derived from the combination of the adiabatic condition \(PV^\gamma = \ ext{constant}\) and the ideal gas law, applying a relationship between temperature and volume.

Evaluating Option (c)

The expression \(T^{\top} P^{1-\gamma} = \ ext{constant}\) is another form for adiabatic condition incorporating pressure and temperature. While it can be rearranged from the relationship between \(PV^\gamma\) and reshuffling with the ideal gas law, the expression given here might be stated incorrectly due to typographical reasons.

Concluding Step

All of the options describe forms of equations derived from the adiabatic process. (c) had a possibility of typographical inconsistency, yet theoretically is meant as a derived form. Thus, answer all options collectively.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle that ties together the pressure, volume, and temperature of an ideal gas. Represented by the equation \( PV = nRT \), where:

• \( P \) stands for pressure,
• \( V \) indicates volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature in Kelvin.

This equation provides an excellent approximation for the behavior of many gases under typical conditions. An ideal gas is a theoretical gas where particles have no volume and do not interact with one another, which simplifies calculations and lets us focus on the parameters of pressure, volume, and temperature. In the context of adiabatic processes, the Ideal Gas Law becomes a backbone for other derived equations, even though no heat exchange occurs.

Heat Capacity Ratio

The heat capacity ratio, denoted as \( \gamma \), is the ratio of the heat capacity at constant pressure, \( C_p \), to the heat capacity at constant volume, \( C_v \).\[ \gamma = \frac{C_p}{C_v} \]This ratio is crucial in describing adiabatic processes. It determines how the pressure, volume, and temperature of an ideal gas change when the gas undergoes adiabatic expansion or compression. In an adiabatic process, no heat is exchanged with the surroundings, making the heat capacity ratio a key player in determining how much the gas compresses or expands without external energy input.- For monatomic gases, \( \gamma \approx 5/3 \).- For diatomic gases, \( \gamma \approx 7/5 \).The value of \( \gamma \) tells us about the degrees of freedom of gas molecules, essentially how energy is distributed during the process.

Pressure-Volume Relationship

One of the hallmarks of adiabatic processes is the specific pressure-volume relationship encapsulated in the equation \( PV^{\gamma} = \text{constant} \). This shows that, during any adiabatic expansion or compression, the product of pressure (\( P \)) and the volume (\( V \)) raised to the power of \( \gamma \) remains constant.- If the volume increases, the pressure decreases proportionally to maintain the equation.- Conversely, if the volume decreases, the pressure must increase to keep the relation constant.This equation is derived from the first law of thermodynamics, assuming no heat exchange occurs with the environment. It's a pivotal concept in understanding how real gases behave under rapid expansion or compression.

Temperature-Volume Relationship

The temperature-volume relationship for an adiabatic process in an ideal gas is expressed with the equation \( TV^{\gamma-1} = \text{constant} \). This tells us how temperature changes as the volume of gas changes, without transferring heat to or from the surroundings.- When the volume increases during an adiabatic expansion, the temperature of the gas decreases.- Conversely, as the volume decreases during compression, the temperature rises.Just as with pressure-volume relationships, this is derived considering the preservation of energy in the absence of heat transfer. The meticulous balance outlined by this equation helps predict the state changes in systems not exchanging heat, capturing how tightly interlinked volume and temperature remain.