1. The Clausius-Clapeyron Equation, Study notes of Thermodynamics

Download 1. The Clausius-Clapeyron Equation and more Study notes Thermodynamics in PDF only on Docsity! 1. The Clausius-Clapeyron Equation We will utilize the Carnot cycle to derive an important relationship, known as the Clausius-Clapeyron Equation or the first latent heat equation. This equation describes how saturated vapor pressure above a liquid changes with temperature and also how the melting point of a solid changes with pressure. Let the working substance in the cylinder of a Carnot ideal heat engine be a liquid in equilibrium with its saturated vapor and let the initial state of the substance be T− and es, Leg1-2 Let the cylinder be placed on a source of heat at temperature T and let the substance expand isothermally until a unit mass of the liquid evaporates. In this transformation the pressure remains constant at es, and the substance passes from state 1 to 2. If the specific volumes of liquid and vapor at temperature T are αl and αv, respectively, the increase in the volume of the system in passing from 1 to 2 is (αv − αl). Also the heat absorbed from the source is Lv where Lv is the latent heat of vaporization. Leg2-3 The cylinder is now placed on a nonconducting stand and a small adi- abatic expansion is carried out from 2 to 3 in which the temperature falls from T to T − dT and the pressure from es − des. Leg3-4 The cylinder is placed on the heat sink at temperature T − dT and an isothermal and isobaric compression is carried out from state 3 to 4 during which vapor is condensed. Leg4-1 We finalize by an adiabatic compression from es − des and T − dT to es and T . All the transformations are reversible, so We can define the efficiency as in the Carnot Cycle η = w qh = qh − qc qh = Th − Tc Th (1) And in this specific case of an infinitesimal cycle We can define the efficiency as in the Carnot Cycle dw qh = dT T (2) 1 The work done in the cycle is equal to the area enclosed on a p− V diagram. Therefore dw = (αv − αl)des (3) Also, qh = lv, therefore, lv T = (αv − αl)des dT (4) Which can be re-written as des dT = lv T (αv − αl) (5) Which is the Clausius-Clapeyron Equation 1a. Proof of Clausius-Clapeyron using Gibbs Function or Gibbs Free Energy For any two phases (1 and 2) in equilibrium g1 = g2 (6) (7) Proof: In equilibrium T and P of both phases are equal. There is no NET transfer of mass, dg1 = 0 and dg2 = 0. Now, if there is a change of temperature from T by dT and the corresponding change in pressure from P by dP and liquid is vaporizing. The Gibbs free energy of the vapor will change by the same amount as that of the liquid. dg1 = dg2 (8) (9) applying the fundamental relations: −(s2 − s1)dT + (α2 − α1)dp = 0 (10) or 2 pressures at -12.5◦C (Figure 2 and Figure 3). Figure 1: From Bohren and Albrecht ”This difference has profound consequences. Cloud droplets freeze at tem- peratures below 0◦C when they have the help of ice nunclei to initiate freezing. Nature has many cloud condensation nuclei, but very few ice nuclei, consequently clouds of supercooled water droplets are the rule rather than the exception. The cloud may be mostly sub cooled water droplets, but if one droplet happens to form on an ice nucleus it would have an advantage over its neighbors. Evap- oration proceeds more slowly from an ice crystal than from a wager droplet at the same temperature, and hence an ice crystal in an environment of mostly sub cooled water droplets grows at the expense of its neighbors shrinking (Figure 4). As the ice grain grows, it falls faster than its smaller neighbors, therefore colliding with them. They freeze upon contact (riming) making the ice grain even larger. As it consumes cub cooled water droplets on its descent, it falls ever faster with an ever larger cross-sectional area, both of which serve to increase its ability to grow even larger. Thus the rich (ice grains) get richer and the poor (sub cooled water droplets) get poorer. This is the Bergeron Process. ” From Bohren and Albrecht. 5 10 Pres oo a - 2 | Subcooled Water Saturation Vapor Pressure (mb) Figure 5.6 Suluration vapor pressure of ice and subcooled water (pure water existing as lig- uid al a temperature below the 0 nominal freezing point). Temperature (~C) o “30 -25 -20 -15 -10 -5 0 Figure 2: From Bohren and Albrecht (e,-e,) (mb) 04. = . aa a 0.0 be a \ | Figure §.7 Difference between -30 -25 the saturation vapor pressure of 20-15 - 0656 20 -18 10-8 subcooled waler e, and thal of ice ¢,. Note the maximum at Temperature (°C) thon. 1c Figure 3: From Bohren and Albrecht