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equipartition

# equipartition - Each degree of freedom of a system has an...

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Specific heat of ideal gases and the equipartition theorem Specific heats revisited The specific heat of a material will be different depending on whether the measurement is made at constant volume or constant pressure. Molar specific heat at constant volume c V (1/n) dQ/dT| V But if dV = 0 then dW = 0 and by the first law of thermodynamics, dE int = dQ , c V = (1/n) E int / T | V true for all materials . Specializing to ideal gases, we know E int (T) c V = (1/n) dE int /dT or dE int = nc V dT true for all ideal gases .

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Molar specific heat at constant pressure c P (1/n) dQ/dT| P and using the first law, c P = (1/n) [dE int +dW]/dT| P = (1/n)dE int /dT| P + (1/n)dW/dT| P For an ideal gas, E int (T) and dW| P = d(PV)| P = P dV| P = P d(nRT/P)| P = nRdT c P = c V + R true for all ideal gases . Equipartition Theorem
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Unformatted text preview: Each degree of freedom of a system has an energy ½ k B T. A degree of freedom is an independent mode of motion: translation, rotation, and vibration. Let the number of degrees of freedom be denoted f. Neglecting vibration, for ideal gases : N=nN A and R=N A k B 1/ndE/dT c V +R System f E int (T) c V c P m.i.g. 3N 3N ½ k B T 3/2 R 5/2 R (trans.) 3/2 nRT d.i.g. 5N 5N ½ k B T 5/2 R 7/2 R dumbbell(trans.+rot.) 5/2 nRT p.i.g. 6N 6N ½ k B T 6/2 R 4 R (trans.+rot.) 6/2 nRT Ideal Gas Toolkit Ideal gas law PV = nRT First Law of Thermo. dE int = dQ in- dW out Processes isothermal, adiabatic, etc. Work by gas W out = ∫p dV Internal energy of gas E int (T) = f ½ k B T EXAMPLES [in class]...
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