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Chapter 2: Problem 12

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

Short Answer

Expert verified
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

01

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: \[ \Delta E_{total} = 0 \]
02

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: \[ E_{total} = U + K + V \]
03

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: \[ \Delta U + \Delta K + \Delta V = 0 \]
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 鈥� 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鈥檚 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.

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Most popular questions from this chapter

A heat pump cycle whose coefficient of performance is \(2.5\) delivers energy by heat transfer to a dwelling at a rate of \(20 \mathrm{~kW}\). (a) Determine the net power required to operate the heat pump, in \(\mathrm{kW}\). (b) Evaluating electricity at \(\$ 0.08\) per \(\mathrm{kW} \cdot \mathrm{h}\), determine the cost of electricity in a month when the heat pump operates for 200 hours.

The drag force, \(F_{\mathrm{d}}\), imposed by the surrounding air on a vehicle moving with velocity \(\mathrm{V}\) is given by $$ F_{\mathrm{d}}=C_{\mathrm{d}} \mathrm{A}_{2}^{\frac{1}{2}} \rho \mathrm{V}^{2} $$ where \(C_{\mathrm{d}}\) is a constant called the drag coefficient, \(\mathrm{A}\) is the projected frontal area of the vehicle, and \(\rho\) is the air density. Determine the power, in \(\mathrm{kW}\), required to overcome aerodynamic drag for a truck moving at \(110 \mathrm{~km} / \mathrm{h}\), if \(C_{\mathrm{d}}=0.65, \mathrm{~A}=10 \mathrm{~m}^{2}\) and \(\rho=1.1 \mathrm{~kg} / \mathrm{m}^{3}\).

A flat surface is covered with insulation with a thermal conductivity of \(0.08 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The temperature at the interface between the surface and the insulation is \(300^{\circ} \mathrm{C}\). The outside of the insulation is exposed to air at \(30^{\circ} \mathrm{C}\), and the heat transfer coefficient for convection between the insulation and the air is \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Ignoring radiation, determine the minimum thickness of insulation, in \(\mathrm{m}\), such that the outside of the insulation is no hotter than \(60^{\circ} \mathrm{C}\) at steady state.

Air undergoes two processes in series: Process 1-2: polytropic compression, with \(n=1.3\), from \(p_{1}=\) \(100 \mathrm{kPa}, v_{1}=0.04 \mathrm{~m}^{3} / \mathrm{kg}\) to \(v_{2}=0.02 \mathrm{~m}^{3} / \mathrm{kg}\) Process 2-3: constant-pressure process to \(v_{3}=v_{1}\) Sketch the processes on a \(p-v\) diagram and determine the work per unit mass of air, in \(\mathrm{kJ} / \mathrm{kg}\).

A gas is compressed from \(V_{1}=0.3 \mathrm{~m}^{3}, p_{1}=1\) bar to \(V_{2}=0.1 \mathrm{~m}^{3}, p_{2}=3\) bar. Pressure and volume are related linearly during the process. For the gas, find the work, in \(\mathrm{kJ}\).
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