Question

Consider an airplane trip from Chicago, Illinois, to Denver, Colorado. List some path-dependent functions and some state functions for the plane trip.

Short answer

Path-dependent functions for an airplane trip from Chicago to Denver include total fuel consumption, flight distance, time taken, flying altitude, and the number of turns taken during the journey. These variables depend on the specific route taken by the airplane. On the other hand, state functions, such as the passengers' location, initial and final temperature, altitude at departure and arrival, aircraft's initial and final weight, and total potential energy change, depend only on the initial and final states of the trip and are not influenced by the particular path taken.

Step-by-step solution

Path-Dependent Functions

1. Total fuel consumption: The amount of fuel used by the airplane depends on the specific route it takes. A longer route will consume more fuel than a shorter one. 2. Flight distance: The distance traveled by the airplane depends on the path it takes from one city to another. A longer path will have a greater flight distance. 3. Time taken: The time it takes for the airplane to travel between the two cities depends on the route taken, as well as factors like air traffic, stopovers, and the speed of the airplane. 4. Flying altitude: The altitude at which the airplane flies can vary depending on the route and weather conditions. 5. Number of turns: The number of turns required to navigate from the origin to the destination depends on the specific path taken by the airplane.

State Functions

1. Passengers' location: The plane's passengers will start in Chicago and end in Denver, regardless of the path taken. 2. Initial and final temperature: The temperature inside the airplane cabin at the start and end of the flight depend on the weather in Chicago and Denver and is independent of the path taken. 3. Altitude at departure and arrival: The altitude of the airports in Chicago and Denver are state functions since they are properties of the airports and are not influenced by the flight path of the airplane. 4. Aircraft's initial and final weight: The weight of the airplane at the beginning and end of the flight includes variables like fuel, cargo, and passengers, and it will be the same regardless of the path taken. 5. Total potential energy change: The change in potential energy from Chicago to Denver depends only on the difference in altitude between the two cities, which is independent of the path taken during the trip.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The fundamental principles of thermodynamics are encapsulated in its four laws, which explain how energy is conserved and transferred.

Chemical systems, like the fuel used in an airplane, abide by these laws. In the context of a plane trip, a state function in thermodynamics might include the internal energy of the system, which is independent of how the system arrived at that state. Conversely, the total heat produced during the flight, which is path-dependent, would not be a state function as it depends on the specific conditions such as efficiency, speed, and altitude of the plane. The study of thermodynamics in such scenarios helps in understanding how to optimize fuel consumption and manage the plane's energy efficiently.

Chemical Systems

A chemical system encompasses substances and reactions undergoing chemical changes. In the context of aviation, the combustion of airline fuel, a chemical reaction, powers the aircraft. During the chemical process, energy is released, which propels the airplane forward.

In thermodynamics, the state of a chemical system can be described using various state functions such as enthalpy, entropy, and Gibbs free energy. These quantities are significant for understanding and predicting the direction and spontaneity of chemical reactions. For instance, the enthalpy change of the fuel during the flight is a state function as it solely depends on the initial and final states of the fuel, not on the path it undergoes in the airplane's engines.

Energy Conservation

Energy conservation is a principle stating that energy cannot be created or destroyed in an isolated system, only transformed from one form into another. This is encapsulated in the first law of thermodynamics, also known as the law of energy conservation.

During a plane trip, potential energy and kinetic energy are continually converted back and forth. The fuel's chemical energy is converted into kinetic energy to move the airplane and potential energy as the airplane gains altitude. The state function of total potential energy change between Chicago and Denver does not depend on the journey's path but on the altitude difference between the two locations, demonstrating energy conservation at different points in the trip.