According to statistical mechanics, the translation energy of a system of point particles is given by 3/2 NKT. And it is known that a single particle exhibits only translational energy. So can we simply imply that single particle system energy can be obtained just by substituting N=1? Because as far as I remember, the first principle of statistical mechanics assumes that number of particles of a system is extremely large, so we can't directly apply those principle for a single particle system
Yes 1/2 kT per degree of freedom. Science should decide whether k is a fundamental property of nature or just a convenient conversion factor. Tolman treated it as invariant conversion factor in describing relativistic thermodynamics. Others have treated it as a fundamental property of nature. Committees vote first one way and then the other.
Yes, it is called the eqipartition theorem, you have 1/2 kT per degree of freedom
or 3/2 kT per simple particle, if you have 3 dimensions for translation, or 3/2 NkT
for N simple particles (for the energy) and yes it is meant for large systems.
If you have a single particle as an isolated system, talking about temperature makes no much sense, since we have different ways to define it, and they all differ if you have only 1 particle (for further details, see "Microcanonical Ensemble").
But, if the single particle is in thermic equilibrium with a reservoir at a temperature T, you can treat its statistcs with Canonical Ensemble, and them you will find that its energy is 3/2 KT.
Even though the canonical ensemble can be used to treat few particles, it assumes that they are in thermal equilibrium with a reservoir which is itself a system of many particles. So the principle of statistical mechanics still holds true.
Marcos and Moon Bakaya Hazarika,
A single particle has no state. That means, by detecting a single particle one cannot say in which state it is. If a single particle is PREPARED according to a procedure known to you as a pure state, then you can confirm the state by testing the particle. If the preparation procedure prepares a mixture of states, one cannot be sure that measuring the single particle and finding that it corresponds to one of the states in the mixture, that was the state of the particle before the measurement. This issue is under debate.
As to the particles in the gas, they have different energies. 3/2KT is one of the global estimations - there exists an average energy, a most probable energy, you can find details in Wikipedia.
Yes 1/2 kT per degree of freedom. Science should decide whether k is a fundamental property of nature or just a convenient conversion factor. Tolman treated it as invariant conversion factor in describing relativistic thermodynamics. Others have treated it as a fundamental property of nature. Committees vote first one way and then the other.
To which principle do you refer when talking of the 1st principle of statistical mechanics? Here are the two basic principles of the statistical mechanics:
1. The complete state of the mechanical system is mathematically encoded, at a given time, as a phase point (classical mechanics), or a pure quantum state-vector (quantum mechanics).
2. An equation of motion carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (non-relativistic quantum mechanics), or Dirac, or Klein-Gordon (relativistic quantum mechanics).
Of course we can't say that the translation energy of a single particle is 3/2 KBT, because this is the average energy in the Boltzmann distribution of energy. The energy of a single particle can take any value, with a probability given by the distribution.
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