Clausius Clapeyron Equation

Size: px

Start display at page:

Download "Clausius Clapeyron Equation"

Deirdre Ellis
5 years ago
Views:

1 Course - BSc. Applied Physical Science (Computer Science) Year & Semester - Ist, IInd Subject - Physics Paper No. - 6 Paper Title - Thermal Physics Lecture No. 18 Clausius Clapeyron Equation Hello friends, after having a detailed look at the Maxwell s relations in the earlier lecture and learning that they have far-reaching implications in thermodynamics and are frequently used to derive useful thermodynamic relations we now move on to the next important topic the Clapeyron equation. The Clapeyron equation is one such relation which enables us to determine the enthalpy change associated with a phase change (such as the enthalpy of vaporization) from a knowledge of P, v, and T data alone. Consider the third Maxwell relation, During a phase-change process, the pressure is the saturation pressure, which depends on the temperature only and is independent of the specific volume. That is, P!"# = f T!"#. Therefore, the partial derivative ( P/ T )! can be expressed as a total derivative (dp/dt)!"#, which is the slope of the saturation curve on a P-T diagram at a specified saturation state.

2 This slope is independent of the specific volume, and thus it can be treated as a constant during the integration of third Maxwell equation between two saturation states at the same temperature. For an isothermal liquid vapor phase-change process, for example, the integration yields Or A During this process the pressure also remains constant. Therefore, from equation We get Substituting this result into Equation A, we obtain Which is called the Clapeyron equation after the French engineer and physicist E. Clapeyron. This is an important thermodynamic relation since it enables us to determine the enthalpy of vaporization h!! at a given temperature by simply measuring the slope of the saturation curve on

3 a P-T diagram and the specific volume of saturated liquid and saturated vapor at the given temperature. The Clapeyron equation is applicable to any phase-change process that occurs at constant temperature and pressure. It can be expressed in a general form as Here the subscripts 1 and 2 indicate the two phases. The Clapeyron equation can be simplified for liquid vapor and solid vapor phase changes by utilizing some approximations. At low pressures v! v! and thus v!! v!. By treating the vapor as an ideal gas, we have v! = RT P. Substituting these approximations into Clapeyron equation, we find Or For small temperature intervals h!! can be treated as a constant at some average value. Then integrating this equation between two saturation states yields This equation is called the Clapeyron Clausius equation, and it can be used to determine the variation of saturation pressure with temperature. It can also be used in the solid vapor region by replacing h!! by h!!, i.e. the enthalpy of sublimation, of the substance. General Relations for du, dh, and ds,

4 The state postulate established that the state of a simple compressible system is completely specified by two independent, intensive properties. Therefore, at least theoretically, we should be able to calculate all the properties of a system at any state once two independent, intensive properties are available. This is certainly good news for properties that cannot be measured directly such as internal energy, enthalpy, and entropy. However, the calculation of these properties from measurable ones depends on the availability of simple and accurate relations between the two groups. In this discussion we develop general relations for changes in internal energy, enthalpy, and entropy in terms of pressure, specific volume, temperature, and specific heats alone. We shall also develop some general relations involving specific heats. The relations developed will enable us to determine the changes in these properties. The property values at specified states can be determined only after the selection of a reference state, the choice of which is quite arbitrary. Internal Energy Changes We choose the internal energy to be a function of T and v; that is, u = u(t, v) and taking its total differential we get, Using the definition of c v, we have A Now we choose the entropy to be a function of T and v; that is, s = s(t, v) and taking its total differential we get, Substituting this into the T ds relation du = Tds Pdv yields Equating the coefficients of dt and dv in equation A and B gives..b

5 Using the third Maxwell relation, we get Substituting this into Equation A, we obtain the desired relation for du as: The change in internal energy of a simple compressible system associated with a change of state from ( T!, v! ) to (T!, v! ) is determined by integration of: Enthalpy Changes The general relation for dh is determined in exactly the same manner. This time we choose the enthalpy to be a function of T and P, that is, h = h(t, P), and taking its total differential we get,

6 Using the definition of c!, we have...c Now we choose the entropy to be a function of T and P; that is, we take s = s(t, P) and take its total differential as, Substituting this into the T ds relation dh = T ds + v dp gives..d Equating the coefficients of dt and dp in equation C and D, we obtain Using the fourth Maxwell relation, We have Substituting this into equation C, we obtain the desired relation for dh as:

7 The change in enthalpy of a simple compressible system associated with a change of state from (T1, P1) to (T2, P2) is determined by integration as: The Joule-Thomson Coefficient When a fluid passes through a restriction such as a porous plug, a capillary tube, or an ordinary valve, its pressure decreases. We have seen in lectures that, the enthalpy of the fluid remains approximately constant during such a throttling process. You will remember that a fluid may experience a large drop in its temperature as a result of throttling, which forms the basis of operation for refrigerators and air conditioners. This is not always the case, however. The temperature of the fluid may remain unchanged, or it may even increase during a throttling process. The temperature behavior of a fluid during a throttling process is described by the Joule- Thomson coefficient, defined as μ = T P! Thus the Joule-Thomson coefficient is a measure of the change in temperature with pressure during a constant-enthalpy process. Notice that, if Joule-Thomson coefficient is

8 during a throttling process. A careful look at its defining equation reveals that the Joule-Thomson coefficient represents the slope of h = constant lines on a T-P diagram. Such diagrams can be easily constructed from temperature and pressure measurements alone during throttling processes. A fluid at a fixed temperature and pressure T1 and P1 (thus fixed enthalpy) is forced to flow through a porous plug, and its temperature and pressure downstream are measured. The experiment is repeated for different sizes of porous plugs, each giving a different set of temperature and pressure readings. Plotting the temperatures against the pressures gives us a h = constant line on a T-P diagram.

9 Repeating the experiment for different sets of inlet pressure and temperature and plotting the results, we can construct a T-P diagram for a substance with several h = constant lines. Some constant-enthalpy lines on the T-P diagram pass through a point of zero slope or zero Joule-Thomson coefficient. The line that passes through these points is called the inversion line, and the temperature at a point where a constant-enthalpy line intersects the inversion line is called the inversion temperature. The temperature at the intersection of the P = 0 line and the upper part of the inversion line is called the maximum inversion temperature. Notice that the slopes of the h = constant lines are negative at states to the right of the inversion line and positive to the left of the inversion line. A throttling process proceeds along a constant-enthalpy line in the direction of decreasing pressure, that is, from right to left. Therefore, the temperature of a fluid increases during a throttling process that takes place on the right-hand side of the inversion line. However, the fluid temperature decreases during a throttling process that takes place on the left-hand side of the inversion line. It is clear from this diagram that a cooling effect cannot be achieved by throttling unless the fluid is below its maximum inversion temperature. This presents a problem for substances whose maximum inversion temperature is well below room temperature. For hydrogen, for example, the maximum inversion temperature is 68 C. Thus hydrogen must be cooled below this temperature if any further cooling is to be achieved by throttling. So friends here we come to the end of our discussion therefore we sum up: The Clapeyron equation enables us to determine the enthalpy change associated with a phase change from knowledge of P, v, and T data alone. The temperature behavior of a fluid during a throttling process is described by the Joule- Thomson coefficient. The Joule-Thomson coefficient is a measure of the change in temperature of a substance with pressure during a constant-enthalpy process. So friends, that is it for today. See you in the next lecture with more on thermodynamics. Thank you very much.

Chapter 12 PROPERTY RELATIONS. Department of Mechanical Engineering

Chapter 12 PROPERTY RELATIONS. Department of Mechanical Engineering Chapter 12 THERMODYNAMIC PROPERTY RELATIONS Dr Ali Jawarneh Department of Mechanical Engineering Hashemite University it Objectives Develop fundamental relations between commonly encountered thermodynamic