Equipartition of Energy
The number of degrees of freedom can be defined as the minimum number of independent coordinates, which can specify the configuration of the system completely. (A degree of freedom of a system is a formal description of a parameter that contributes to the state of a physical system.)
The position of a rigid body in space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.
The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have:
• For a single particle in a plane two coordinates define its location so it has two degrees of freedom; • A single particle in space requires three coordinates so it has three degrees of freedom; • Two particles in space have a combined six degrees of freedom; • If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five, because the distance formula can be used to solve for the remaining coordinate once the other five are specified.
The equipartition theorem relates the temperature of a system with its average energies. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in the translational motion of a molecule should equal that of its rotational motions.
At temperature T, the average energy of any quadratic degree of freedom is 1/2 kT.
For a system of N molecules, each with f degrees of freedom: 1 U = N f kT thermal · · 2 The difficulty is to count the number of degrees of freedom f.
Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of 3N independent simple harmonic oscillators, where N denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy kT, the average total energy of the solid is 3NkT, and its heat capacity is 3Nk. Heat Heat Q is energy transferred from one body to another by thermal conduction (by molecular contact), electromagnetic radiation, or convection (bulk motion of a gas or liquid) – caused by a difference in temperature
Simulation of thermal convection. Red hues designate hot areas, while regions with blue hues are cold. A hot, less-dense lower boundary layer sends plumes of hot material upwards, and likewise, cold material from the top moves downwards. This illustration is taken from a model of convection in the Earth's mantle. The first law of thermodynamics
In thermodynamics, work W performed by a closed system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. If the total energy inside the system (internal energy) is U, then