# Adsorption of Ammonium, Nitrate, and Phosphate on Hydrochars and Biochars
## 3 Results and Discussion
### Adsorption Thermodynamics  
In the design of sorption systems, two categories of thermodynamic properties are essential: directly measurable properties, like temperature and the equilibrium constant, and properties that cannot be directly measured, such as entropy (\(S\)) and free energy (\(G\)). To determine the thermodynamic properties that cannot be directly measured, the well-known thermodynamic relation Gibbs free energy (\(G\)) is employed. Both Gibbs free energy and entropy play a crucial role in determining the characteristics of sorption chemical reactions or the overall nature of the sorption process. Evaluating these thermodynamic parameters is instrumental in gaining insights into the nature of the adsorption process [50]. The differential heat of adsorption was calculated using the Clausius-Clapeyron expression in the form  
\[\Delta H=-2.303R\left(\frac{T_{1}T_{2}}{T_{1}-T_{2}}\right)logC_{2}-logC_{1} \tag{4}\]  
where \(C_{2}\) and \(C_{1}\) are the equilibrium concentrations at temperatures \(T_{2}\) and \(T_{1}\), respectively. For each adsorption isotherm, we obtained \(C_{2}\) and \(C_{1}\) at a fixed \(q_{\rm e}\) value using the Langmuir isotherm expression. Then, the obtained \(C_{2}\) and \(C_{1}\) values were substituted in Equation (4) to obtain the differential heat of adsorption. Equation (5) can be used to obtain the adsorption isosteric heat, which is a positive quantity and is given by the following relationship:  
\[q_{\rm st}=-\Delta H \tag{5}\]  
By employing other thermodynamic parameters, including the Gibbs free energy, \(\Delta G\), and the entropy change, \(\Delta S\) was obtained from the following thermodynamic relation:  
\[T\Delta S=\Delta H-\Delta G \tag{6}\]  
The free energy change can be calculated using the expression  
\[\Delta G=RT\ \rm{ln}C_{\rm e}/C_{\rm s} \tag{7}\]  
where \(C_{\rm s}\) is the solubility of the solute, and it depends on the adsorption temperature. The free energy change obtained using Equation (7) is independent of the temperature within the range of temperatures studied. Figure 6a-i show the variation in the free energy change, change in the heat content, and entropy change for the transfer of the solutes (single components) ammonium, nitrate, and phosphate from the bulk solution to the adsorbed state on the adsorbent surface of BC700, HC180, and the AC at 50 \({}^{\circ}\)C for all the studied systems. Irrespective of the solute and adsorbent type, we observe a general trend for the variation in the thermodynamic parameters as a function of the amount adsorbed. The Gibbs free energy increases with coverage. This can be expected as the equilibrium concentration increases with the adsorption at a fixed T. The differential heat of adsorption decreases with coverage or adsorption, \(q_{\rm e}\). At lower \(q_{\rm e}\), the adsorption is dominated by solute-solid interactions; however, as the adsorption or the coverage increases, the adsorption is dictated by both solute-solute and solute-solid interactions, and this explains the decrease in the \(\Delta H\) value with the increase  
in \(q_{\rm e}\). The solid-solute interactions at lower surface adsorption can be due to the adsorption onto the active sites with the highest binding affinity; these typically arise from pores of a size that allows them to commensurate with solute ions or the functional groups. For the case of entropy, it clearly decreases with the increase in the surface coverage. This is an expected observation as the system moves from a disordered state to a more ordered adsorbed state. From the differential heat of adsorption, we calculate the adsorption isosteric heat at zero coverage by estimating that \(q_{\rm st}=-\Delta H\) at \(q_{\rm e}=0\). The calculated \(q_{\rm st}\) values at zero coverage for BC700, HC180, and the AC are plotted as a function of the solute. In Figure 7, it can be seen that HC180 has strong adsorption towards nitrate ions.