We present fully traceable two-parameter Huckel equations (with parameters B and b(1)) for the activity coefficient of sodium chloride and for the osmotic coefficient of water in aqueous NaCl solutions at temperatures from (0 to 80) degrees C. These equations apply within experimental error to all thermodynamic data available for these solutions at least up a molality of 0.2 mol.kg(-1). In our previous study (J. Chem. Eng. Data 2016, 61, 286-306), these equations were successfully tested against the literature results of electrochemical, isopiestic, and cryoscopic measurements usually in the temperature range from (0 to 25) degrees C. There, a constant value was employed for B, whereas a linear model with respect to the temperature was utilized for b(1). The linear model was determined from the values of b(1) at 0 degrees C and at 25 degrees C obtained from freezing-point depression data and from isopiestic and cell-potential difference data, respectively. In the present study, these two b(1) values are utilized alongside the constant value of parameter B but a new quadratic model is presented for the temperature dependence of b(1). The third data point required for this model is obtained froth the direct vapor pressure measurements of Gibbard et al. (J. Chem. Eng. Data 1974, 19, 281-288) at 75 degrees C. The results obtained with this quadratic equation for b(1) agree well with the test results of the linear model in the previous paper (see the citation above) up to 25 degrees C. The most important new test results above that temperature are reported here. Our quadratic model has additionally been tested with all the high-precision calorimetric data available in the literature for NaCl solutions. In this first part (Part 1) of the study, the test results from the thermodynamic quantities associated with partial molar enthalpy are reported. In the forthcoming second part (Part 2) of the study, the results of the quantities associated with the heat capacity of NaCl solutions will be considered. In the tests of these two parts, all calculations dealing with calorimetric data are performed in a new way. Both the calorimetric data and the vapor pressure data (from both direct and isopiestic measurements) can be predicted using the new Huckel equations within experimental error in dilute NaCl solutions from (0 to 80) degrees C. For comparison, also other Huckel models are considered and at best these apply up to the molality of the saturated NaCl solution at various temperatures. Following the success of the new models, new values for the activity coefficients, osmotic coefficients, relative apparent molar enthalpies, and relative partial molar enthalpies for NaCl solutions at rounded molalities are reported at the end of this Article. We have good reasons to believe that the new values contain the most reliable ones available for the given thermodynamic quantities.