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Equipartition of Energy 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 ∆U = Q + W (energy conservation for a thermodynamic system). The first explicit statement of the first law of thermodynamics, by Rudolf Clausius in 1850, referred to cyclic thermodynamic processes. "In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.” Rudolf Julius Emanuel Clausius (1822 –1888) Compression work Work done by the gas is: W = PA∆s = P ∆V Work done on the gas (assuming equistatic compression) is: W = P ∆V − Vf W = P (V )dV − ZVi Isothermal compression: P = NkT/V V W = NkT ln i Vf Q = ∆U W = W (heat leaves the gas) − − An adiabatic process is any process occurring without gain or loss of heat within a system (i.e. during the process the system is thermodynamically isolated- there is no heat transfer with the surroundings). A key concept in thermodynamics, many rapid chemical and physical processes are described or approximated in this way. Q =0, ∆U = W P = NkT/V f dU = NkdT = P dV 2 − f dT dV = VTf/2 = const 2 T − V PVγ = const γ =(f + 2)/f (adiabatic exponent) Heat capacity Heat capacity C is the measurable physical quantity that shows the amount of heat required to change a substance's temperature by a given amount. Specific heat capacity c is the heat capacity per unit mass. Q ∆U W C C = − c = ⌘ ∆T ∆T m Heat capacity at constant volume (W=0): ∆U @U CV = = ∆T V @T V ✓ ◆ ✓ ◆ Heat capacity at constant pressure: ∆U + P ∆V @U @V CP = = + P ∆T P @T P @T P ✓ ◆ ✓ ◆ ✓ ◆ If the system stores thermal energy only in “quadratic” degrees of freedom: @ NfkT Nfk CV = = @T 2 2 … if f does not depend on T… ✓ ◆ + vibration + rotation CV /R translation Idealized plot of the molar specific heat of a diatomic gas against temperature. It agrees with the value (7/2)R predicted by equipartition at high temperatures (where R is the gas constant), but decreases to (5/2)R and then (3/2)R at lower temperatures, as the vibrational and rotational modes of motion are "frozen out". The failure of the equipartition theorem led to a paradox that was only resolved by quantum mechanics. For most molecules, the transitional temperature Trot is much less than room temperature, whereas Tvib can be ten times larger or more. A typical example is carbon monoxide, CO, for which Trot ≈ 2.8 K and Tvib ≈ 3103 K. For molecules with very large or weakly bound atoms, Tvib can be close to room temperature (about 300 K). For solids: rule of Dulong and Petit (1819) C /R 3 V ! For the ideal gas: @V @ NkT Nk = = @T @T P P ✓ ◆P ✓ ◆ CP = CV + Nk = CV + Rn.
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