We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé...We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815;2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .展开更多
Using a space filled with black-body radiation, we derive a generalization for the Clausius-Clapeyron relation to account for a phase transition, which in-volves a change in spatial dimension. We consider phase transi...Using a space filled with black-body radiation, we derive a generalization for the Clausius-Clapeyron relation to account for a phase transition, which in-volves a change in spatial dimension. We consider phase transitions from dimension of space, n, to dimension of space, (n - 1), and vice versa, from (n - 1) to n -dimensional space. For the former we can calculate a specific release of latent heat, a decrease in entropy, and a change in volume. For the latter, we derive an expression for the absorption of heat, the increase in entropy, and the difference in volume. Total energy is conserved in this transformation process. We apply this model to black-body radiation in the early universe and find that for a transition from n = 4 to (n - 1) = 3, there is an immense decrease in entropy accompanied by a tremendous change in volume, much like condensation. However, unlike condensation, the volume change is not three-dimensional. The volume changes from V4, a four-dimensional construct, to V3, a three-dimensional entity, which can be considered a subspace of V4. As a specific example of how the equation works, we consider a transition temperature of 3 × 1027 Kelvin, and assume, furthermore, that the latent heat release in three-dimensional space is 1.8 × 1094 Joules. We find that for this transition, the internal energy densities, the entropy densities, and the volumes assume the following values (photons only). In four-dimensional space, we obtain, u4 = 1.15×10125 J? m-4, s4 = 4.81×1097 J? m-4? K-1, and V4 = 2.14×10-31 m4. In three-dimensional space, we have u3 = 6.13×1094 J? m-3, s3 = 2.72×1067 J? m-3? K-1, and V3 = 0.267 m3. The subscripts 3 and 4 refer to three-dimensional and four-dimensional quantities, respectively. We speculate, based on the tremendous change in volume, the explosive release of latent heat, and the magnitudes of the other quantities calculated, that this type of transition might have a connection to inflation. With this work, we prove that space, in and of itself, has an inherent energy content. This is so because giving up space releases latent heat, and buying space costs latent heat, which we can quantify. This is in addition to the energy contained within that space due to radiation. We can determine the specific amount of heat exchanged in transitioning between different spatial dimensions with our generalized Clausius-Clapeyron equation.展开更多
A partial phase diagram characterizing the conformational change that occurs in Thermomyces lanuginosus xylanase as it is slowly heated in 150 mM sodium phosphate (pH = 7.0) has been con-structed from slow-scan-rate d...A partial phase diagram characterizing the conformational change that occurs in Thermomyces lanuginosus xylanase as it is slowly heated in 150 mM sodium phosphate (pH = 7.0) has been con-structed from slow-scan-rate differential scanning calorimetry measurements. The Clausius-Clapeyron equation was applied to determine an associated volume change of -205 L·mol-1 at 24°C, the equilibrium transition temperature at 1.0 atm pressure. This value is in excellent agreement with that predicted using a previously published [1] empirical equation for calculating the hydro-dynamic radius if the transition is regarded as from a random coil to a functional, folded state and with the assumption that the hydrodynamic radius is a good approximation of the true random coil radius. The existence of a low-temperature random coil is confirmed by circular dichroism and dynamic light scattering measurements. Thus, at 24°C and 1.0 atm pressure the enzyme appears to fold from a random coil to a functional, folded form as it is slowly heated.展开更多
文摘We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815;2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .
文摘Using a space filled with black-body radiation, we derive a generalization for the Clausius-Clapeyron relation to account for a phase transition, which in-volves a change in spatial dimension. We consider phase transitions from dimension of space, n, to dimension of space, (n - 1), and vice versa, from (n - 1) to n -dimensional space. For the former we can calculate a specific release of latent heat, a decrease in entropy, and a change in volume. For the latter, we derive an expression for the absorption of heat, the increase in entropy, and the difference in volume. Total energy is conserved in this transformation process. We apply this model to black-body radiation in the early universe and find that for a transition from n = 4 to (n - 1) = 3, there is an immense decrease in entropy accompanied by a tremendous change in volume, much like condensation. However, unlike condensation, the volume change is not three-dimensional. The volume changes from V4, a four-dimensional construct, to V3, a three-dimensional entity, which can be considered a subspace of V4. As a specific example of how the equation works, we consider a transition temperature of 3 × 1027 Kelvin, and assume, furthermore, that the latent heat release in three-dimensional space is 1.8 × 1094 Joules. We find that for this transition, the internal energy densities, the entropy densities, and the volumes assume the following values (photons only). In four-dimensional space, we obtain, u4 = 1.15×10125 J? m-4, s4 = 4.81×1097 J? m-4? K-1, and V4 = 2.14×10-31 m4. In three-dimensional space, we have u3 = 6.13×1094 J? m-3, s3 = 2.72×1067 J? m-3? K-1, and V3 = 0.267 m3. The subscripts 3 and 4 refer to three-dimensional and four-dimensional quantities, respectively. We speculate, based on the tremendous change in volume, the explosive release of latent heat, and the magnitudes of the other quantities calculated, that this type of transition might have a connection to inflation. With this work, we prove that space, in and of itself, has an inherent energy content. This is so because giving up space releases latent heat, and buying space costs latent heat, which we can quantify. This is in addition to the energy contained within that space due to radiation. We can determine the specific amount of heat exchanged in transitioning between different spatial dimensions with our generalized Clausius-Clapeyron equation.
文摘A partial phase diagram characterizing the conformational change that occurs in Thermomyces lanuginosus xylanase as it is slowly heated in 150 mM sodium phosphate (pH = 7.0) has been con-structed from slow-scan-rate differential scanning calorimetry measurements. The Clausius-Clapeyron equation was applied to determine an associated volume change of -205 L·mol-1 at 24°C, the equilibrium transition temperature at 1.0 atm pressure. This value is in excellent agreement with that predicted using a previously published [1] empirical equation for calculating the hydro-dynamic radius if the transition is regarded as from a random coil to a functional, folded state and with the assumption that the hydrodynamic radius is a good approximation of the true random coil radius. The existence of a low-temperature random coil is confirmed by circular dichroism and dynamic light scattering measurements. Thus, at 24°C and 1.0 atm pressure the enzyme appears to fold from a random coil to a functional, folded form as it is slowly heated.
