A More General View of Temperature
When a high temperature object is placed in contact with a low temperature object, then energy will flow from the high temperature object to the lower temperature object, and they will approach an equilibrium temperature. When the details of this common-sense scenario are examined, it becomes evident that the simple view of temperature embodied in the commonly used kinetic temperature approach has some significant problems.
The above illustration summarizes the situation when the kinetic temperature gives a reasonable general description of the nature of temperature. For monoatomic gases acting like point masses, a higher temperature simply implies higher average kinetic energy. Faster molecules striking slower ones at the boundary in elastic collisions will increase the velocity of the slower ones and decrease the velocity of the faster ones, transferring energy from the higher temperature to the lower temperature region. With time, the molecules in the two regions approach the same average kinetic energy (same temperature) and in this condition of thermal equilibrium there is no longer any net transfer of energy from one object to the other.
Complications such as these have led to the adoption of a different approach to the concept of temperature in the study of thermodynamics. Schroeder's proposal for a theoretical definition of temperature is:
The kinetic temperature for monoatomic ideal gases described above is consistent with this definition of temperature for the simple systems to which it applies. In that case the equilibrium reached is one of maximum entropy, and the rate of approach to that state will be proportional to the difference in temperature between the two parts of the system. Noting that the equilibrium state of a collection of particles will be the state of greatest multiplicity, then one can define the temperature in terms of that multiplicity (entropy) as follows:
Temperature is expressed as the inverse of the rate of change of entropy with internal energy, with volume V and number of particles N held constant. This is certainly not as intuitive as molecular kinetic energy, but in thermodynamic applications it is more reliable and more general.
The Relationship of Entropy to Temperature
The definition of temperature in terms of molecular kinetic energy, the "kinetic temperature", is commonly used in introductory treatments of thermodynamics. But in view of practical difficulties with that approach, temperature is often defined in terms of two other state variables, entropy S and internal energy U.
Temperature is expressed as the inverse of the rate of change of entropy with internal energy, with volume V and number of particles N held constant. This is certainly not as intuitive as molecular kinetic energy, but in thermodynamic applications it is more reliable and more general. This form of the temperature can be obtained from the thermodynamic identity.
To apply this definition of temperature to a monoatomic ideal gas, we need an expression for the entropy of an ideal gas:
Then making use of the definition of temperature in terms of entropy:
This gives an expression for internal energy that is consistent with equipartition of energy.
with kT/2 of energy for each degree of freedom for each atom.
Another example of this approach to temperature is the Einstein solid with q units of energy and N oscillators. The internal energy U can be represented by q times the oscillator energy unit hf = ε. The entropy can be expressed as
Taking the derivative gives
This is what is expected from equipartition of energy. Each oscillator has two degrees of freedom, and each should represent kT/2 of energy, giving U=NkT.