20.5:
Internal Energy
The internal energy of a system can be defined as the sum of the kinetic and potential energies of all the individual atoms in the system.
Consider a system of a monoatomic ideal gas. When the gas is heated at constant volume, the kinetic energy of the atoms increases, but since there is no interaction between the atoms, their potential energy remains zero.
Therefore, the internal energy depends on the average kinetic energy of the monoatomic atoms. It is expressed as three over two NkBT, where N is the number of gas atoms, kB is the Boltzmann constant, and T is its temperature. Thus, the internal energy of an ideal gas at constant volume depends only on temperature.
The internal energy of a system is a state function. Boiling water that has been cooled to room temperature has the same internal energy as water obtained by melting ice to room temperature.
Thus, the internal energy depends only on the state of the system, not on the path employed to obtain it. Therefore, it is path-independent.
20.5:
Internal Energy
The internal energy of a thermodynamic system is the sum of the kinetic and potential energies of all the molecules or entities in the system. The kinetic energy of an individual molecule includes contributions due to its rotation and vibration, as well as its translational energy. The potential energy is associated only with the interactions between one molecule and the other molecules of the system. Neither the system's location nor its motion is of any consequence as far as the internal energy is concerned.
Consider an ideal monatomic gas. Here, each molecule is a single atom. Consequently, there is no rotational or vibrational kinetic energy. Furthermore, there are no interatomic interactions, so potential energy is assumed to be zero. The internal energy is, therefore, a result of translational kinetic energy only. Therefore, the internal energy of ideal gas is just the number of molecules multiplied by the average kinetic energy per molecule. Thus, for n moles of an ideal monatomic gas, the internal energy is given by
It can be seen that the internal energy of a given quantity of an ideal monatomic gas depends on the temperature and is independent of the pressure and volume of the gas. For other systems, the internal energy cannot be expressed so simply. However, an increase in internal energy can often be associated with an increase in temperature.
In general, when a quantity of heat Q is added to a system, and the system does no work during the process, the internal energy increases by an amount equal to Q. When a system does work W by expanding against its surroundings, and no heat is added during the process, energy leaves the system and the internal energy decreases. While Q and W depend on the path, the change in internal energy of a system during any thermodynamic process depends only on the initial and final states, not on the path leading from one to the other.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. p. 629.
- OpenStax. (2019). University Physics Vol. 2. [Web version] Pp. 115–116. Retrieved from https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy