1 Experiment 3. Heat-Capacity Ratios for Gases. Adiabatic Expansion
Experiment 3. Heat-Capacity Ratios for Gases. Adiabatic Expansion Method Background For the reversible adiabatic expansion of a perfect gas, the change in energy content is related to the change in volume by nRT dE = − pdV = − dV = −nRTd lnV (4) V Since E for a perfect gas is a function of temperature only, we can also write dE = CvdT. Substituting this expression into Eq. (4) and integrating, we find that ~ ~ T2 V2 Cv ln = −R ln ~ (5) T1 V1 ~ ~ where Cv and V are molar quantities (that is, Cv/n, V/n). We assumed here that Cv is constant over the temperature range involved. Eq. (5) predicts the decrease in temperature resulting from reversible adiabatic expansion of a perfect gas. In experiment, we will use the following two-step process involving a perfect gas denoted by A: Step I: Allow the gas to expand adiabatically and reversibly until the pressure has dropped from p1 to p2: ~ ~ A(p1 ,V1 ,T1 )→ A(p2 ,V2 ,T2 ) (6) Step II: At constant volume, restore the temperature of the gas to T1: ~ ~ A(p2 ,V2 ,T2 )→ A(p3 ,V2 ,T1 ) (7) For step I, we can use the perfect-gas law to obtain ~ T2 p2V2 = ~ (8) T1 p1V1 ~ ~ Substituting Eq. (8) into Eq. (5) and combining terms in V2 V1 , we can write ~ ~ ~ C ~ p2 − (Cv + R) V2 p V2 ln = ~ ln ~ = − ~ ln ~ (9) p1 Cv V1 Cv V1 since for a perfect gas ~ ~ C p = Cv + R (10) 1 For step II, which restores the temperature to T1, ~ V2 p1 ~ = (11) V1 p3 Thus ~ p1 C p p1 ln = ~ ln (12) p2 Cv p3 This can be rewritten in the form ~ C p ln(p1 p2 ) γ ≡ ~ = (13) Cv ln(p1 p3 ) The goal of our experiment is to obtain γ for two gases, N2 and Ar, through measurements of p1, p2, and p3.
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