Question

When an ideal gas expands at a constant temperature, \(\Delta E=0\) for the change. Why?

Short answer

For an ideal gas during an isothermal expansion, the internal energy change is zero (\(\Delta E=0\)) because the internal energy is a function of temperature, which remains constant. Any heat energy added to the system is used to perform work, with no change in internal energy.

Step-by-step solution

Understanding the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy of a system, \(\Delta E\), is equal to the heat added to the system, \(Q\), minus the work done by the system, \(W\): \[\Delta E = Q - W\]. For an ideal gas expanding at constant temperature (isothermal expansion), \(\Delta E\) represents the change in internal energy.

Relating Change in Internal Energy to Temperature

For an ideal gas, the internal energy is only a function of temperature. In an isothermal process, since the temperature (\(T\)) is constant, the internal energy of the gas does not change. Hence, \(\Delta E = 0\).

Applying the First Law of Thermodynamics to an Isothermal Process

As \(\Delta E = 0\) during an isothermal expansion, by rearranging the first law, we obtain \[0 = Q - W\], which implies that the heat added to the system (\(Q\)) is equal to the work done by the system (\(W\)). Thus, any heat input is converted entirely into work without changing the internal energy.

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics is a fundamental principle in physics that relates to energy conservation. It essentially states that energy cannot be created or destroyed in an isolated system. The law can be expressed in the context of a thermodynamic system through the equation \[\Delta E = Q - W\].
In this equation, \(\Delta E\) represents the change in internal energy of the system, \(Q\) is the heat added to the system, and \(W\) is the work done by the system on its surroundings. One of the implications of this law is the concept of energy transfer: when a system undergoes any process, energy must be conserved, it can only be transferred in or out of the system in the form of heat or work.

For an isothermal expansion of an ideal gas, where the temperature remains constant, the internal energy (\(\Delta E\)) remains unchanged. This means that any heat energy (\(Q\)) added into the system must be completely transformed into work (\(W\)), facilitating an expansion under constant temperature conditions. This principle helps us understand heat engines, refrigerators, and even the basic principles governing our atmosphere.

Internal Energy

Internal energy is a concept central to understanding thermodynamic systems and denotes the total energy contained within a system. It is the sum of the potential and kinetic energy of the particles that compose the system. In the context of an ideal gas, the internal energy is directly related to the temperature of the gas. As outlined by the relation \[ U = \frac{3}{2}nRT \],
where \(U\) is the internal energy, \(n\) the number of moles, \(R\) the ideal gas constant, and \(T\) the absolute temperature in Kelvin. Because of this direct relationship, in an isothermal process where the temperature is constant, there is no change in the internal energy (\(\Delta E = 0\)).

The internal energy remains the same regardless of the amount of work done or heat transferred. In our daily lives, when we heat water on the stove (non-isothermal), the temperature and thus the internal energy of the water increase; however, when we leave a balloon filled with air outside on a warm day, the air inside expands isothermally with constant internal energy.

Ideal Gas Behavior

The ideal gas behavior is a simplified model that describes the properties and behavior of gases under various conditions. According to this model, an ideal gas is composed of particles that have no volume and do not exert forces onto each other, except during elastic collisions. The ideal gas law can be given as \[ PV = nRT \],
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the absolute temperature.

During an isothermal expansion, the temperature (\(T\)) of an ideal gas remains constant. As the gas expands and volume (\(V\)) increases, the pressure (\(P\)) of the gas must decrease to maintain the equation's balance, assuming the number of moles (\(n\)) does not change. This expansion is one of the most crucial aspects of gas behavior and can be used to explain natural phenomena such as the rising of a hot air balloon or the functioning of an internal combustion engine. Understanding the ideal gas law is essential for comprehending the underlying mechanisms of many technologies and natural processes.