# Calorimetric Measurements of Biological Interactions and Their Relationships to Finite Time Thermodynamics Parameters

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## Abstract

**:**

## 1. Introduction

## 2. Calorimetry

#### 2.1. Background

_{REF}and P

_{SAM}are the power to the reference and sample cell, respectively. ${\dot{E}}_{REL}$ equals the rate of energy released by the reactions occurring in the sample cell. The energy balance on the sample cell provides additional insights into these assumptions and the mechanisms that occur during the reaction, Equation (2).

_{SAM}and c

_{s}equal the mass of material and the specific heat of the material in the sample cell, T

_{s}is the temperature of the sample cell at any time and T

_{R}is the reference temperature for the internal energy. The mass of material, ${m}_{sam}$, in a typical titration experiment is related to the injection rate, concentration, and injecting time. $\dot{{m}_{inj}}$ is the injection of the syringe in Equation (4). A similar energy balance can be written for the reference cell where there is no rate of energy released term. Since the temperature of both cells is controlled to be equal in the theory of two-cells ITC and they are geometrically similar, the heat loss from both cells will be the same at any time. The internal energy balance in the reference cell is constant as its mass does not vary with time. Since the temperature of each cell is controlled to a constant value, the temperature time derivatives are also zero. The implications of these assumptions will be discussed later. Combining these energy balances of sample and reference cell under these conditions yields the following relationship:

_{sam}, if any, and the heat transfer characteristics of the cell, ${q}_{L,s}$. A calibration relation is required to determine these heat transfer characteristics. The energy balance on this device is similar to that of Equation (2) and can be solved for energy released, ${\dot{E}}_{REL}$. The integration of the energy release term over the time of the process can be used to determine the enthalpy of reaction as stated above.

_{HT}A).

_{D}. The equilibrium constant allows the introduction of second law parameters such as the Gibbs free energy change of the reaction, which then allows the entropy of the system during the reaction to be determined. A second law analysis of the calorimeter is then used to calculate the entropy production rate, which is related to the irreversibility of the reaction (Section 2.3). Procedures and hypotheses concerning the relationships for energy stored in growth, growth rate, and generation times would need to be applied to possibly formulate entropy statements for these processes, similar to that carried out in the ITC analysis [10].

#### 2.2. Case 1: Spatially Uniform System Heating without Reaction

_{∞}. The energy balance is developed as in Equation (5).

_{∞}

_{S}= 4180 kJ/(kg K), thermal conductivity = 0.6 W/(m K), and density of 997 kg/m

^{3}. P

_{SAM}is set to zero in this simulation. The volume of the liquid is fixed at 80 µL and the (u

_{HT}A) product is 0.0356 W/K. The value of the power into the cell is zero in this experiment. Equation (9) was solved numerically using the code developed by Modaresifar and Kowalski [11] for reactingmixtures in microchambers, which simulates an injection experiment and analyzes its thermal process in a microscale calorimeter. This is a straightforward problem that is easily solved. This code was found to spatially converge to an accurate solution with three nodes for this small volume and uniform heat generation rate. It predicts the temperature time response, as shown in Figure 2, until the steady state is reached, approximately t

_{f}= 130 s. In Figure 2 the temperatures of the three spatial nodes are shown. They are not distinguishable from one another, which confirms that the numerical simulation satisfies the uniformly spatial assumption. Tests at a higher number of nodes, providing more precise resolution in the simulation, confirm that the solution has converged.

_{g}, in Equation (9). This calculation is accomplished by integrating Equation (9) with respect to time under the assumption that q

_{g}is constant. In this step, note that one is specifying the function form of the internal heat generation, a constant, that is occurring in the sample, i.e., a mechanism. The integrated form of Equation (9) is:

_{L,s}, in Equation (12) is the heat transfer rate and is summarized in Figure 3 as a function of time using Equation (10). As expected, the heat transfer rate starts at zero and increases to its steady value. The difference between the heat transfer rate and the rate of energy released from the heat generation term is the internal energy storage rate, the first term on the RHS of Equation (12).

#### 2.3. Case 2: Spatially Uniform System with Reaction Limited by an Equilibrium Constant

_{B}and the products are as shown in Equation (17), which assumes a complete reaction.

_{SAM}= 0 is:

_{TOT}]. During this injection, the rate of reaction is considered to be very fast and the rate of energy released per volume, q

_{g2}, is not a constant and is determined from the equilibrium constant k

_{B}and the enthalpy of reaction per mol of the injected compound, ΔH. The rate of energy release between two specified times is:

_{TOT}, is known at the start and the injection rate of compound [X] is held constant. Unlike case 1, the rate of energy added to the chamber is not constant but is a function of time that starts at a maximum value and then decreases to zero, or a constant value as the number of reaction sites or available food goes towards zero, Figure 7.

_{B}. The thermodynamics of the equilibrium composition K

_{B}is related to the change in the Gibbs energy.

_{B}would then be used in Equation (30) to determine the change in the Gibbs Energy and the change in the absolute entropy property change. These results, together with Equation (26), would allow one to isolate the rate of entropy productions and measure the irreversibility of the reaction.

## 3. Discussion: What Is Measured

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic sketches of two calorimeter configurations used in determining the energy released from biological/reacting systems. (

**A**) is a two-cell configuration where both cells are usually maintained at a uniform temperature and the difference between the power of the reference and sample cell is recorded. (

**B**) is a single cell configuration where the temperature is monitored during the biological reaction as a function of time.

**Figure 2.**The temperature response of the single sample cell as a function of time for a pure material, water, exposed to a uniform heat generation rate.

**Figure 3.**Instantaneous heat transfer rate as a function of time for the sample cell. The results shown are based on the known value of the heat transfer coefficient, (u

_{HT}A) product, and the observed temperature from Figure 2 as if it were a calorimeter experiment.

**Figure 4.**Summary of the rate of entropy flow out of the sample cell as a function of time. The calculations are based on temperatures and heat losses which would be determined from calorimetric measurements.

**Figure 5.**Summary of the rate of entropy property changes within the sample cell as a function of time. The calculations are based on temperatures and heat losses which would be determined from calorimetric measurements.

**Figure 6.**The rate of entropy production within the sample cell as a function of time. The calculations are based on temperatures and heat losses which would be determined from calorimetric measurements.

**Figure 7.**Comparison of the constant energy release rate of case 1 to the time dependent energy release rate of case 2, the reacting compound example. The orange line is the constant energy release of case 1. The blue monotonically decreasing line is for case 2.

**Figure 8.**Predicted temperature vs. time response for the reacting calorimeter cell. This response corresponds to the observed parameter of the calorimeter experiment.

**Figure 9.**Predicted heat flow rate vs. time response for the reacting calorimeter cell. This parameter is calculated using the predicted temperature response and the heat transfer characteristics of the test cell as determined from a calibration test.

**Figure 10.**Predicted entropy flow rate out vs. time response for the reacting calorimeter cell. This parameter is calculated using the predicted temperature response and heat transfer rate.

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Zhang, Y.; Kowalski, G.J.
Calorimetric Measurements of Biological Interactions and Their Relationships to Finite Time Thermodynamics Parameters. *Entropy* **2022**, *24*, 561.
https://doi.org/10.3390/e24040561

**AMA Style**

Zhang Y, Kowalski GJ.
Calorimetric Measurements of Biological Interactions and Their Relationships to Finite Time Thermodynamics Parameters. *Entropy*. 2022; 24(4):561.
https://doi.org/10.3390/e24040561

**Chicago/Turabian Style**

Zhang, Yuwei, and Gregory J. Kowalski.
2022. "Calorimetric Measurements of Biological Interactions and Their Relationships to Finite Time Thermodynamics Parameters" *Entropy* 24, no. 4: 561.
https://doi.org/10.3390/e24040561