Question

Is it possible to compress an ideal gas isothermally in an adiabatic piston- cylinder device? Explain.

Short answer

Answer: No, it is not possible to compress an ideal gas isothermally in an adiabatic piston-cylinder device, as the conditions for isothermal and adiabatic processes lead to a contradiction in the internal energy change (ΔU) of the system.

Step-by-step solution

Writing the Ideal Gas Law

The Ideal Gas Law relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of moles (n) and the ideal gas constant (R): PV = nRT In our case, since the process is isothermal, the temperature remains constant (T = constant).

Considering the First Law of Thermodynamics for ideal gases

The First Law of Thermodynamics states that the internal energy change (ΔU) of a system equals the heat added to the system (Q) minus the work done by the system (W): ΔU = Q - W Now, let's consider the conditions of our problem: the process is adiabatic (Q = 0) and the gas is being compressed, i.e., work is being done on the system (W > 0). With these conditions, our equation becomes: ΔU = -W As we know, for an ideal gas, internal energy (U) is proportional to its temperature (T): ΔU = nCvΔT, where Cv is the specific heat at constant volume. Since T is constant in an isothermal process (ΔT = 0), we have that the internal energy change (ΔU) must be equal to zero: ΔU = nCv × 0 = 0

Evaluation of the adiabatic and isothermal conditions

From the previous steps, we concluded that in an isothermal process, the internal energy change must be equal to zero (ΔU = 0). However, recall that in our adiabatic process, internal energy change is equal to the negative work done on the system (ΔU = -W). Now, in our problem, the gas is being compressed, which means that work is done on the gas (W > 0), implying that ΔU < 0. This contradiction between the isothermal and adiabatic conditions shows that it is not possible to compress an ideal gas isothermally in an adiabatic piston-cylinder device.

Key concepts

Ideal Gas Law

When studying gases in physics and chemistry, one of the most fundamental equations is the Ideal Gas Law, represented by the formula \( PV = nRT \). This relationship tells us that the pressure \( P \) multiplied by the volume \( V \) of a gas equals the number of moles \( n \) of the gas multiplied by the universal gas constant \( R \) and the temperature \( T \), which is measured in Kelvin.

Understanding this law is crucial because it allows us to predict how a gas will respond to changes in pressure, volume, and temperature, as long as it behaves ideally. This means that the particles of the gas must not attract or repel each other significantly, and the volume they occupy is negligible compared to the volume of the container.

Applying the Ideal Gas Law in Real-world Situations

When working on exercises like gas compression, one should consider that while the Ideal Gas Law provides a good approximation, real gases show deviations from ideal behavior under high-pressure and low-temperature conditions.

First Law of Thermodynamics

The First Law of Thermodynamics is a version of the law of conservation of energy. This principle asserts that the change in the internal energy of a system \( (\Delta U) \) is equal to the heat \( Q \) added to the system minus the work \( W \) done by the system \( (\Delta U = Q - W) \).

In simple terms, this means that energy can neither be created nor destroyed—it can only be transferred or changed from one form to another. When we compress a gas, for example, we are doing work on the system which potentially changes its internal energy, depending on whether heat is added or removed.

Conservation of Energy in Practice

Energy conservation is fundamental in understanding thermodynamic processes. Whether we're heating a home or powering an engine, the First Law of Thermodynamics tells us that we need to account for where the energy comes from and where it goes.

Isothermal Process

An isothermal process is a thermodynamic change where the temperature remains constant—\( T = \text{constant} \). This means that all energy added to a system or work done on a system during an isothermal process takes the form of heat transfer because the internal energy of an ideal gas is solely dependent on its temperature. If we remember that a real gas behaves ideally at higher temperatures and lower pressures, we can predict the outcomes of several industrial and natural procedures that involve gases.

The difficulty in achieving a true isothermal process in practice is that it requires very slow changes or perfect thermal contact with an environment that can absorb or release heat without changing its own temperature.

Real-Life Isothermal Examples

Isothermal conditions are often assumed for calculations because they simplify the complexity of real-life systems by negating the need to factor in temperature changes.

Adiabatic Process

An adiabatic process occurs when a system does not exchange heat with its surroundings (\( Q = 0 \)). Instead, all changes in internal energy are due to work done on or by the system. When a gas is compressed adiabatically, it typically increases in temperature because the work done on the gas increases its internal energy, leading to a rise in temperature.

Adiabatic processes are also reversible in an ideal world, meaning that if the process were reversed, the system would return to its original state without any heat flow. This concept is important in areas such as meteorology and the study of engines, where rapid changes occur that approximate adiabatic conditions.

Reversibility in Adiabatic Processes

Even though perfect adiabatic conditions are hard to achieve in reality, understanding these processes helps engineers design more efficient systems by minimizing heat loss.

Internal Energy

The internal energy of an ideal gas is the total kinetic energy of its atoms or molecules in motion, which is related to the temperature of the gas. According to the First Law of Thermodynamics, the internal energy change (\( \Delta U \)) can be caused by adding or removing heat (\( Q \)), doing work on or by the system (\( W \)), or a combination of both.

In the context of ideal gases, if the temperature is constant (as in an isothermal process), the internal energy does not change (\( \Delta U = 0 \)), since it is only temperature dependent. This principle is key in predicting how a gas responds to different physical situations and in creating models for energy transfer.

Importance of Internal Energy in Thermodynamics

Understanding internal energy is essential in the study of thermodynamics as it underpins concepts like heat capacity, enthalpy, and the various thermodynamic processes that are foundational to physics and engineering applications.