### **Specified Physicochemical Properties**
For systematic and quantitative classification, the most relevant properties available from the data include:
1. **Polarity (as inferred mostly from TPSA and XLogP):**
    - *TPSA* (Topological Polar Surface Area): Directly correlates with the ability of a molecule to participate in hydrogen bonding. Suitable for comparing overall molecular polarity.
    - *XLogP*: Partition coefficient (logP) quantitatively ranks hydrophobicity/hydrophilicity. Negative values = hydrophilic, positive values = hydrophobic.
    - *H-bond Donor/Acceptor Count.*: Useful secondary metric if values cluster.

2. **Hydrogen bonding capacity (sum of H-bond donor and acceptor counts):**
    - Higher counts generally indicate more "protic" or polar solvents.

3. **Ambiguity for Multi-component/Misnamed:** Substances like "MeOH/H2O_V2 9:1" and "THF_V2" lack quantitative database values and will be separately addressed.

---

### **Summary Table of Key Data**

| Substance     | TPSA   | XLogP | HBD | HBA | Classification Variables     |
|---------------|--------|-------|-----|-----|-----------------------------|
| Methanol (MeOH) | 20.2 | -0.5  | 1   | 1   | Highly polar, hydrophilic   |
| DMF           | 20.3   | -1.0  | 0   | 1   | Polar, hydrophilic, aprotic |
| MeCN          | 23.8   | 0.0   | 0   | 1   | Polar, aprotic              |
| THF           | 9.2    | 0.5   | 0   | 1   | Moderately polar, aprotic   |

*HBD = H-bond donor count; HBA = H-bond acceptor count*

---

### **Data-driven Classification**

#### **Step 1: Polarity (TPSA & XLogP)**
- Polarity scale (TPSA): Typically TPSA < 20 = less polar/aprotic; 20–40 = moderate/strong polarity.
- Hydrophobicity (XLogP): Negative = hydrophilic; near zero = borderline; positive = hydrophobic.

#### **Step 2: Hydrogen Bonding Capability**
- **Protic:** HBD ≥ 1 (can donate H-bonds)
- **Aprotic:** HBD = 0 (cannot donate H-bonds); potential for HBA only

---

### **Quantitative Grouping:**
#### **Group 1: Highly Polar Protic Solvent**
- **Methanol (MeOH)**
    - TPSA: 20.2; XLogP: -0.5; HBD = 1 → Protic, highly polar, hydrophilic

#### **Group 2: Polar Aprotic Solvents**
- **DMF, MeCN**
    - DMF (TPSA 20.3, XLogP -1.0, HBD 0)
    - MeCN (TPSA 23.8, XLogP 0.0, HBD 0)
    - Quantitative data separates them from MeOH (protic), but both cluster as polar/aprotic.

#### **Group 3: Moderately Polar Aprotic Solvent**
- **THF**
    - TPSA: 9.2 (markedly lower than others), XLogP: 0.5, HBD = 0
    - Appropriately its own cluster by quantitative polarity score.

#### **Unable to Classify: Mixture/Missing Data**
- **MeOH/H2O_V2 9:1, THF_V2**
    - No direct chemical property data in the database.
    - *Cannot assign to a quantitative class based on the presented values!*

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## **Final Classification Table**

| Group                       | Members            | Quantitative Justification                                                                              |
|-----------------------------|--------------------|---------------------------------------------------------------------------------------------------------|
| 1. Highly Polar Protic      | MeOH               | TPSA 20.2, XLogP -0.5, HBD 1                                                                            |
| 2. Polar Aprotic            | DMF, MeCN          | TPSA 20.3–23.8, XLogP ~0, HBD 0                                                                         |
| 3. Moderately Polar Aprotic | THF                | TPSA 9.2 (distinctly lower), XLogP 0.5, HBD 0                                                           |
| Unclassified                | MeOH/H2O_V2 9:1, THF_V2 | No database value; composition not clearly defined (mixture/misnamed, so excluded from quantitative groupings) |

---

### **Justification for Number of Groups**
Based strictly on quantitative data, **three groups** are justified:
- **(1) Protic/polar**, 
- **(2) Polar/aprotic**, 
- **(3) Moderately polar/aprotic.**

Further clustering (such as combining all into one 'polar organic solvents' group) would erase the quantitative distinctions revealed by XLogP, TPSA, and HBD that support three distinct clusters.

---

### **Summary**
**Methanol** is a highly polar protic solvent (unique in this set).  
**DMF and MeCN** are polar aprotic solvents, with very similar quantitative profiles.  
**THF** is a moderately polar aprotic solvent, quantitatively distinct by lower TPSA.  
**Mixtures or uncharacterized entries remain unclassified** due to lack of direct quantitative data.

**This trichotomous scheme is fully supported by the provided quantitative data.**