Systems, States and Energy

Introduction: The first step of any thermodynamics problem is to define a system in which the energy will be analyzed. In addition, knowing what state the matter inside of the system is at is critical to solving the problem.

The Essentials

Systems

Every thermodynamics problem starts by defining the system in which the equations will be applied. A system is the area of space which will be analyzed using the laws of thermodynamics. The boundary of a system is an imaginary line which delineates the system (inside the boundary) and the surroundings (everything else in the universe.)

Importantly, mass and energy can cross the boundary of some systems. A system in which only energy and not mass can pass through the boundary is known as a closed system. A common example of a closed system is a piston-cylinder device or an airtight box. An open system is a system here mass can cross the boundary. A common example of an open system is a turbine or compressor through which air is moving. The mass flow rate of a system, or the time derivative $\frac{dm}{dt}$, is represented by the symbol $\dot{m}$.

States

The state of a system is a way to completely describe the condition of the system. This is done through different properties. The state postulate says that the state of a simple, compressible substance can be fully defined by two independent intensive properties. An intensive property is one that does not depend on the size of the system. Some examples of intensive properties include temperature ( $T$), pressure ( $P$), specific volume ( $\nu$), specific internal energy ( $u$), specific enthalpy ( $h$), and specific entropy ( $s$). Specific properties are used in place of extensive properties and are represented with lowercase letters.

To fully define the state of a system, two independent properties are needed. Most properties are independent the majority of time, however, while a substance is undergoing a phase change, temperature and pressure are dependent on one another. Therefore, an additional property is needed for two phase systems.

Energy

Energy is split into two general categories in thermodynamics; macroscopic and microscopic energies. Macroscopic energies include kinetic and potential energies, while all microscopic energies are summed and represented as the internal energy. The energy of a sustem can be written in either extensive, intensive or rate forms. $$\text{Extensive: }E=KE+PE+U=\frac{1}{2}mv^2+mgz+mu$$ $$\text{Intensive: }e=ke+pe+u= \frac{1}{2}v^2+gz+u$$ $$\text{Rate: }\dot{E}=\dot{m}e=\dot{m}(\frac{1}{2}v^2+gz+u)$$

Energy can be transferred to and from a system through one of three ways: heat transfer ( $Q$), mass transfer ( $\dot{m}$), or work ( $W$). Heat transfer is a change of energy due to a temperature difference. Mass transfer occurs only in open systems, but when mass is moved in or out of the system the overall energy of the system changes. Work is a force applied over some distance. This is often extracted in the form of rotational energy and provided in the form of electric work.

Example

1: An electric vehicle weighs 1600 kg and has a power rating of 110 kW. Find the time needed for the car to accelerate from rest to 100 km/h at full power on a level road.

The first step to solving this problems is to recognize what information is given. The mass, initial velocity, final velocity and rate of work ( $\dot{W}$) are all given. $$m=1600 \text{ km}, v_0=0\text{ m/s}, v_f=100\text{ km/h}, \dot{W}=110\text{ kW}$$ After an understanding of provided information is given, establish any useful equations for solving for the unknown variables. It is useful to start with the variable that is asked for in the problem statement and to work backwards to find all needed information. The problem asks us to find the time elapsed between state one and state two. This can be related to the rate of work in the following way $$\dot{W}=\frac{W}{\Delta t}$$ Since this is a closed system with no heat transfer ( $\dot{m}=Q=0$), the only way for energy to be transferred is through work. This means that the change in energy will be equal to the work done on the system. $$W=\Delta E$$ Applying those equations, we can solve for $\Delta t$ in terms of energy change and rate of work into the system. $$\Delta t=\frac{\Delta E}{\dot{W}}$$ Additionally, the change of energy of a system can be written as the sum of the change of energy from each source ( kinetic, potential, internal). $$\Delta E = \Delta KE + \Delta PE +\Delta U$$ There will be no change to the potential or internal energy of the car, so the only change in energy will come from kinetic energy $$\Delta E = \frac{1}{2}m(v_f^2-v_0^2)$$ Now that all of the variables are defined in terms of given information, the units can be converted to standard units and plugged into the equation. $$v_f=27.7778 \text{ m/s}$$ $$\Delta t=\frac{\frac{1}{2}(1600)(27.778^2)}{110}=617284\text{ J}=617.3\text{ kJ}$$

Practice

1: Water flows in a river with a speed of 10 m/s. Find a) the energy per unit mass, b) the energy contained in 5 kg of water, and c) the energy rate of the water with a flow of 300 kg/s.

2: A car uses an engine that operates using multiple piston-cylinder devices and exhausts heat into the air. If the thermodynamic properties of the engine were to be evaluated, what type of system would be defined and what would be included inside of the system?

3: A pot of water is boiling with a temperature of 102 deg C. Is it possible to know the pressure of the water using only the known temperature? Is the state of the water defined? Why or why not?

Solutions:
1: a) 50 J/kg, b) 250 J, c) 15 kW
2: Closed, the piston-cylinder would be the system and everything else the surroundings.
3: Yes, the pressure can be found because during a phase change, like boiling, temperature and pressure are dependent. The state of the water is not defined because only one independent intensive property is known.

More Resources

This section is where you could put links to other resources you feel might be helpful OpenStax: Comparison Tests