## 3 Results and Discussion
### Theory and Model
Specifically, the deviations between the values predicted by [PERSON]'s condensation theory and those measured experimentally are primarily reflected in two aspects. At external solution concentrations below 0.1 mol\(\cdot\)kg\({}^{-1}\), the predicted values begin to approach a constant greater than 0. Analyzing Equation (13) from a mathematical perspective, as the external solution concentration approaches 0, the mobile salt concentration in the membrane phase will also approach 0. However, the concentration of fixed groups remains relatively stable, leading to \(m_{\rm A}^{\rm m}/m_{\rm s}^{\rm m}\!\!\rightarrow\!\!\infty\). Consequently, the product of the ion activity coefficients in solution ultimately approaches:  
\[\gamma_{+}^{\rm m}\gamma_{-}^{\rm m}\left(m_{\rm NaX}^{\rm s}\to 0\right)= \frac{1}{\xi}\exp\left(-\frac{1}{\xi}\right) \tag{18}\]  
However, the ion activity coefficient in the membrane phase can continue to decrease and approach 0 at low concentrations. Additionally, [PERSON]'s condensation theory cannot account for cases where the ion activity coefficients exceed 1. When [PERSON]'s Counterion Condensation Theory is modeled and \(m_{\rm A}^{\rm m}/m_{\rm s}^{\rm m}\) is plotted as a function of \(\gamma_{+}^{\rm m}\gamma_{-}^{\rm m}\), as shown in Figure 6b, [PERSON]'s predictions fall within the range \((1/\xi)\!\!\exp(-1/\xi)\!\!\!\) to 1. Considering the structural similarities and differences between polyelectrolyte solutions and ion-exchange membranes, two major distinctions arise: (1) [PERSON]'s model assumes that the polyelectrolyte behaves as a fully extended linear chain in localized regions [28], which may not hold for the bulk cross-linked ion-exchange membrane [47; 50]. Consequently, the average projection length between monomers of the polyethylene skeleton cannot simply be taken as 2.5 A when calculating the linear charge density, \(\xi\). (2) In ion-exchange membranes, the line charges are much closer to one another compared to those in polyelectrolyte solutions. Only at high ion concentrations can nearby line charges effectively screen one another. At low ion concentrations, this shielding becomes less effective, resulting in the superposition of multi-segment line charge effects. This leads to a situation where the ion activity coefficient in the membrane surpasses the minimum value predicted by a single-segment line charge model and continues to decrease. Based on the fundamental principles of polymer physics [47], a model that more accurately represents a real polymer chain is proposed. As shown in Figure 7b, this model describes a free imaginary chain consisting of \(n\) bonds of length \(l\), linked by linear connections, with no restrictions on the bond angles \(\theta\) between adjacent bonds or on the internal rotation angles \(\varphi\) of the chain segments [47]. The vector \(\mathbf{r}\) from the head to the tail of the chain can be considered as the sum of \(n\) vectors of length 1 that point in random, unrestricted directions [47]. This problem can be reformulated as the irregular flight problem studied by [PERSON] et al. [47; 62]. To discuss the average projection of \(n\) monomers along the \(X\)-axis, denoted as \(\bar{l}_{x}\), we consider that each bond is equally likely to orient in any direction. Consequently, the projections in all positive directions should statistically equal the projections in all negative directions.