The Joule-Thomson coefficient

The Joule-Thomson coefficient of a gas was defined in Eq. 6.3.3 on page 157 by /xjt = (dT/dp)H- It can be evaluated with measurements of T and p during adiabatic throtthng 

To relate /tji to other properties of the gas, we write the total differential of the enthalpy of a closed, single-phase system in the form 

Next we impose a condition of constant H the ratio dT/ dp becomes a partial derivative  

The left side of this equation is the Joule-Thomson coefficient An expression for the partial derivative (dH/dp)r is given in Table 7.1, and the partial derivative dH/dT)p is the heat capacity at constant pressure (Eq. 5.6.3). These substitutions give us the desired relation 

Thermodynamics and Chemistry, second edWlon, version 3 20)) by Howard DeVoe. Latest version www.chem.umd.9du/therm0b00k 

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To convert the Joule-Thomson coefficient, I, in degrees Celsius per atmosphere to degrees Fahrenheit per atmosphere, multiply by 1.8. 

TABLE 2-149 Additional References Available for the Joule-Thomson Coefficient... 

Since the Joule-Thomson process is isenthalpic, the slope of each line can be represented as (dT/dp)lf. This quantity is referred to as the Joule Thomson coefficient, pj j.. Thus1... 

Values for the Joule-Thomson coefficient can be obtained from equations of state. To do so, one starts with the relationship between exact differentials given by equation (1.37) to write (using molar quantities)... 

Figure 3.7(a) shows experimental values of fj,j T obtained for N gas, while Figure 3.7(b) shows how the Joule-Thomson coefficient for N2 gas changes with pressure and temperature.2... 

Figure 3.7 (a) Joule-Thomson inversion curve (/o.t. = 0) for nitrogen, (b) The Joule -Thomson coefficient of nitrogen gas. At the lowest temperature, 123.15 K. nitrogen liquifies hence the curve for the gas terminates at the vapor pressure. 

These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Thomson coefficient for constant H. 

The Joule-Thomson coefficient p.jx, is positive when a cooling of the gas (a temperature drop) is observed because dP is always negative, p.j x, will be positive when dT is negative. Conversely, p.j x, is a negative quantity when the gas warms on expansion because dT then is a positive quantity. Values of the Joule-Thomson coefficient for argon and nitrogen at several pressures and temperatures are listed in Table 5.5. 

It frequently is necessary to express the Joule-Thomson coefficient in terms of other partial derivatives. Considering the enthalpy as a function of temperature and pressure H T, P), we can write the total differential... 

Joule-Thomson Inversion Temperature. The Joule-Thomson coefficient is a function of temperature and pressure. Figure 5.8 shows the locus of points on a temperature-pressure diagram for which p,jx. is zero. Those points are at the Joule-Thomson inversion temperature 7). It is only inside the envelope of this... 

From Equation (5.68), we know that the pressure coefficient of the molar enthalpy of a gas is related to the Joule-Thomson coefficient p,j x by the equation... 

Because of this relationship between (TT — and p-j x.. the former quantity frequently is referred to as the Joule-Thomson enthalpy. The pressure coefficient of this Joule-Thomson enthalpy change can be calculated from the known values of the Joule-Thomson coefficient and the heat capacity of the gas. Similarly, as (H — is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule-Thomson coefficient. As the fugacity and the Joule-Thomson coefficient are both measures of the deviation of a gas from ideahty, it is not surprising that they are related. 

Although the van der Waals equation is not the best of the semi-empirical equations for predicting quantitatively the PVT behavior of real gases, it does provide excellent qualitative predictions. We have pointed out that the temperature coefficient of the fugacity function is related to the Joule-Thomson coefficient p,j x.- Let us now use the van der Waals equation to calculate p,j.T. from a fugacity equation. We will restrict our discussion to relatively low pressures. 

As Cpm is positive, the sign of the Joule-Thomson coefficient depends on the sign of the expression in parentheses in Equations (10.79) and (10.80). The expression in Equation (10.79) is a quadratic in T, and are two values of T exist at any value of P for which p.j x, = 0. Thus, Equation (10.79) predicts two values of the Joule-Thomson inversion temperature T,- for any pressure low enough for Equation (10.75) to be a good approximation for a. As we saw in Section (5.2) and Figure 5.8, this prediction fits, at least qualitatively, the experimental data for the Joule-Thomson experiment for N2 at low pressure. 

The Joule-Thomson coefficient for a real gas is not zero in the limit of zero pressure ... 

In the period 1852-62, J. P. Joule and W. Thomson (later Lord Kelvin) perfected a clever method for measuring the isenthalpic property (dT/dP)Ih which has come to be called the Joule-Thomson coefficient, symbolized /xJT ... 

The measured Joule-Thomson coefficient /jljt provides valuable information about how the enthalpy of real gases depends on variables other than temperature. To obtain information about the P dependence of H, we can employ the Jacobi cyclic identity (1.14b) to rewrite the Joule-Thomson coefficient as... 

Figure 3.12 Qualitative temperature and pressure dependence of the Joule-Thomson coefficient MotO P) for C02.

Problem Prove that the Joule-Thomson coefficient /jljt satisfies the identity (3.69) Mjt = (TaP — )(V/CP). 

In terms of Table 12.1, T and —P are already standard complementary variables, so the Joule-Thomson coefficient is conveniently evaluated from (12.18), with X=T, Y= -P, Z = H ... 

The Joule-Thomson coefficient is the slope of the isenthalpic lines in the P-T projection. In the region where iJt<0, expansion through the valve (a decrease in pressure) results in an increase in temperature, whereas in the region where pJt >0, expansion results in a reduction in temperature. The latter area is recommendable for applying the PGSS process. 

We conclude that the Joule-Thomson coefficient is a function of both the temperature and the pressure, but, unlike the Joule coefficient, it does not go to zero as the pressure goes to zero. The inversion temperature, the temperature at which fi,T = 0, is also a function of the pressure. The value usually reported in the literature is the limiting value as the pressure goes to zero. 

Some useful relationships may be derived involving the Joule-Thomson coefficient. Let us start with H = H(T, P), to get... 

Thus, the initial and final states of a Joule-Thomson expansion he on a curve of constant enthalpy (isoenthalp) and the Joule-Thomson process occurs at constant enthalpy. The Joule-Thomson coefficient, pJT, is defined as... 

The Joule-Thomson coefficient is the slope of the isoenthalp and is a function of both temperature and pressure. From Eq. (23) and Eq. (7) of Appendix A, we can write... 

Show that the Joule-Thomson coefficient can be obtained from the equation of state of a gas by... 

Thus the change is at constant H, and the Joule-Thomson coefficient is (dT/dP)H. But this can be evaluated easily from our Table of Thermodynamic Relations in Chap. II. It is... 

The Joule Thomson coefficient is the ratio of the temperature decrease to the pressure drop, and is expressed in terms of the thermal expansion coefficient and the heat capacity... 

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