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 kin…
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.
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Q1: What makes up the internal energy of a thermodynamic system?
Internal energy is the sum of kinetic and potential energies of all molecules in a system. Kinetic energy includes translational, rotational, and vibrational contributions, while potential energy arises from interactions between molecules. For an ideal monatomic gas, only translational kinetic energy contributes since atoms have no rotational or vibrational energy and no interatomic interactions exist.
Q2: Why does internal energy of an ideal gas depend only on temperature?
For an ideal monatomic gas, internal energy equals three-halves NkBT, where N is the number of atoms, kB is Boltzmann's constant, and T is temperature. Since there are no interatomic interactions, potential energy is zero. When heated at constant volume, only kinetic energy increases, making temperature the sole determinant of internal energy.
Q3: Is internal energy a path-dependent or path-independent property?
Internal energy is path-independent because it is a state function. Boiling water cooled to room temperature has the same internal energy as water from melted ice at the same temperature. The change in internal energy depends only on initial and final states, not on the thermodynamic path taken between them.
Q4: How does adding heat to a system affect its internal energy?
When heat Q is added to a system and no work is done, internal energy increases by an amount equal to Q. Conversely, when a system does work W by expanding against its surroundings with no heat added, energy leaves the system and internal energy decreases by W. These relationships form the basis of the first law of thermodynamics problem solving.
Q5: What is the difference between internal energy for monatomic and polyatomic gases?
Monatomic ideal gases have internal energy from translational kinetic energy only, expressed as three-halves NkBT. Polyatomic molecules include rotational and vibrational kinetic energy contributions. Additionally, polyatomic systems may have potential energy from intermolecular interactions, making their internal energy more complex and temperature-dependent in different ways.
Q6: Does the location or motion of a system affect its internal energy?
No, neither the system's location nor its bulk motion affects internal energy. Internal energy depends only on the microscopic kinetic and potential energies of molecules within the system. A stationary container and a moving container at the same temperature have identical internal energies regardless of their position or velocity.
Q7: How do pressure and volume changes affect internal energy of an ideal gas?
For an ideal monatomic gas, internal energy depends only on temperature and is independent of pressure and volume. Changes in pressure and volume do not directly alter internal energy unless they cause temperature changes. This independence distinguishes ideal gases from real gases and is fundamental to understanding adiabatic processes for an ideal gas.