Question

What form does the energy balance take for an isolated system? Interpret the expression you obtain.

Short answer

For an isolated system, the energy balance equation is \( \Delta U + \Delta K + \Delta V = 0 \), indicating that the total energy remains constant.

Step-by-step solution

Define the Energy Balance Equation

For an isolated system, no energy can enter or leave the system. The energy balance equation for such a system states that the total energy remains constant over time. Mathematically, this can be expressed as: \[ \Delta E_{total} = 0 \]

Identify Components of Total Energy

The total energy \( E_{total} \) of an isolated system can be broken down into its components which typically include internal energy (\( U \)), kinetic energy (\( K \)), and potential energy (\( V \)). Therefore, we have: \[ E_{total} = U + K + V \]

Apply the Energy Balance Equation to Components

Since the total energy remains constant in an isolated system, the changes in internal energy (\( \Delta U \)), kinetic energy (\( \Delta K \)), and potential energy (\( \Delta V \)) must sum to zero. Thus: \[ \Delta U + \Delta K + \Delta V = 0 \]

Interpret the Result

The expression \( \Delta U + \Delta K + \Delta V = 0 \) indicates that in an isolated system, any increase in one form of energy must be balanced by a corresponding decrease in another form, ensuring that the total energy remains constant.

Key concepts

Isolated System

An isolated system is a concept used in thermodynamics and physics to describe a system that does not exchange energy or matter with its surroundings. This means no heat, work, or mass can cross the system's boundaries.
Here's what that means practically:

• No energy can enter the system.
• No energy can leave the system.
• The system's total energy remains constant.

In an isolated system, the only changes occurring are those that happen internally. This is foundational for understanding how energy behaves within specific confines, making the isolated system an idealized way to study energy conservation.

Total Energy

The total energy in an isolated system is a sum of all types of energy within that system.
The three primary types of energy to consider are:

• Internal Energy (U)
• Kinetic Energy (K)
• Potential Energy (V)

To express the total energy mathematically, we use:
\[ E_{total} = U + K + V \]
This equation captures the idea that the system's total energy is a combination of these different energy forms. Because the system is isolated, the total of these energies stays constant over time.

Internal Energy

Internal energy, denoted by U, is the energy associated with the microscopic components of the system – specifically, the atoms and molecules.
It includes:

• Vibrational energy
• Rotational energy
• Translational energy
• Electronic energy

These are energies due to particles moving and interacting on a molecular scale. Changes in the internal energy can occur due to processes like chemical reactions and phase changes but within an isolated system; these changes must be balanced by changes in kinetic or potential energy.

Kinetic Energy

Kinetic energy (K) is the energy a system has due to the motion of its particles.
The faster the particles move, the higher the kinetic energy will be. Kinetic energy can be calculated using the equation:
\[ K = \frac{1}{2} m v^2 \]
where m is mass and v is velocity.
In an isolated system, if kinetic energy changes, it affects the total energy balance. For example, if there’s an increase in kinetic energy, there must be a decrease in either internal energy, potential energy, or both for the total energy to remain constant.

Potential Energy

Potential energy (V) is the energy stored in the system due to the positions of particles within a force field, like gravity or electromagnetism.
Some common types of potential energy are:

• Gravitational potential energy
• Elastic potential energy
• Electric potential energy

The calculation for gravitational potential energy, for example, is:
\[ V = mgh \]
where m is mass, g is the acceleration due to gravity, and h is height.
In an isolated system, any change in potential energy must be balanced by changes in kinetic or internal energy to maintain the total energy balance. If the potential energy goes up, either kinetic or internal energy must go down, and vice versa.