Let's **classify the candidate solvents** according to key, **quantifiable physicochemical properties** that are typically listed for solvents in scientific databases (e.g., *CRC Handbook of Chemistry and Physics*, *PubChem*, etc.). Common relevant properties include:

1. **Polarity (often dielectric constant, ε, as a quantitative measure)**
2. **Protic/Aprotic nature (quantified by number of hydrogen-bond donor atoms, HBD)**
3. **Boiling Point (°C, indicating volatility)**

Let's compile primary **quantitative data** for each substance:

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## **Quantitative Properties Table**

| Solvent                    | Dielectric Constant (ε) | H-Bond Donors | Boiling Point (°C) | Protic/Aprotic |  
|----------------------------|:----------------------:|:-------------:|:------------------:|:---------------|  
| Methanol (MeOH)            | 32.6                   | 1             | 64.7               | Protic         |  
| DMF (N,N-Dimethylformamide)| 36.7                   | 0             | 153                | Aprotic        |
| MeOH/H₂O (9:1 v/v)         | ≈38*                   | 1-2           | ≈70*               | Protic         |
| THF (Tetrahydrofuran)      | 7.6                    | 0             | 66                 | Aprotic        |
| "THF_V2"                   | 7.6 (assume same)      | 0             | 66                 | Aprotic        |
| MeCN (Acetonitrile)        | 37.5                   | 0             | 81.6               | Aprotic        |

\*Approximate: For 9:1 MeOH/H₂O by volume, the dielectric constant is interpolated between MeOH (32.6) and water (78.4); the HBD count may be slightly higher than pure MeOH.

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## 1. **Polarity (Dielectric Constant ε)**

- **High Polarity:** ε > 30
  - MeOH (32.6), DMF (36.7), MeOH/H₂O (≈38), MeCN (37.5)
- **Low/Medium Polarity:** ε < 15
  - THF (7.6), THF_V2 (7.6)

## 2. **Protic/Aprotic Nature (H-Bond Donors, Functional Group count)**

- **Protic (≥1 H-bond donor):**  
  - MeOH (1), MeOH/H₂O (1-2)
- **Aprotic (no H-bond donors):**  
  - DMF (0), THF (0), THF_V2 (0), MeCN (0)

## 3. **Boiling Point (Volatility, °C)**

- **Low Boiling (<100°C):**  
  - MeOH (64.7), THF (66), THF_V2 (66), MeOH/H₂O (≈70), MeCN (81.6)
- **High Boiling (>100°C):**  
  - DMF (153)

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## **Classification Strategy**

Let's **define three possible groups** based on the above, focusing on the *most discriminating quantitative properties*:

1. **Polarity** (High vs. Low/Medium: based on ε)
2. **Proticity** (HBD presence)
3. **Boiling Point/Volatility** (Low vs. High boiling)

But as instructed, **NO MORE THAN THREE GROUPS**.

### **Optimized Classification**

#### **Group 1: High polarity, protic solvents**
- **Criteria:** ε > 30, ≥1 H-bond donor
- **Members:**  
  - **MeOH**  
  - **MeOH/H₂O**

#### **Group 2: High polarity, aprotic solvents**
- **Criteria:** ε > 30, 0 H-bond donor
- **Members:**  
  - **DMF**  
  - **MeCN**

#### **Group 3: Low polarity, aprotic solvents**
- **Criteria:** ε < 15, 0 H-bond donor
- **Members:**  
  - **THF**  
  - **THF_V2**

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## **Final Classification Table**

| Group                 | Criteria                        | Members                        |
|-----------------------|---------------------------------|-------------------------------|
| 1. High Polarity Protic    | ε > 30, ≥1 HBD                  | MeOH, MeOH/H₂O                |
| 2. High Polarity Aprotic   | ε > 30, 0 HBD                   | DMF, MeCN                     |
| 3. Low Polarity Aprotic    | ε < 15, 0 HBD                   | THF, THF_V2                   |

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### **Justification**

- This grouping is **exclusively based on quantitative data** (dielectric constant, H-bond donor count) directly from standard physicochemical property tables.
- More than three groups is unnecessary, as no solvent straddles categories (no ambiguity based on these discrete values).

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**References for values:**  
- CRC Handbook of Chemistry and Physics  
- PubChem  
- Sigma-Aldrich solvent tables

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**If you need classification based on ONLY ONE property, or want to optimize for a specific use-case (e.g., only polarity), please specify!**