Which of the following statements are correct ?

Options

A. Fullerences have dangling bonds
B. Fullerences are cage-like molecules
C. Graphite is thermodynamically most stable allotrope of carbon
D. Graphite is slippery and hard and therefore used as a dry lubricant in machines.

Explanation

To determine which statements about fullerenes and graphite are correct, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement A: Fullerenes have dangling bonds.**
- Fullerenes are allotropes of carbon, specifically C60, which have a cage-like structure resembling a football. In this structure, the carbon atoms are bonded in a way that does not leave any unbonded or "dangling" bonds. Therefore, this statement is **incorrect**.

2. **Statement B: Fullerenes are cage-like molecules.**
- As mentioned, fullerenes have a distinct cage-like structure formed by a combination of hexagonal and pentagonal arrangements of carbon atoms. This characteristic is a defining feature of fullerenes. Therefore, this statement is **correct**.

3. **Statement C: Graphite is the thermodynamically most stable allotrope of carbon.**
- Under standard conditions, graphite is indeed the most stable allotrope of carbon. This is well-established in thermodynamic studies. Therefore, this statement is **correct**.

4. **Statement D: Graphite is slippery and hard, and therefore used as a dry lubricant in machines.**
- While it is true that graphite is slippery and is used as a dry lubricant, it is **not hard**; in fact, graphite is soft. The hardest allotrope of carbon is diamond. Therefore, this statement is **incorrect**.

### Conclusion:
From the analysis, we conclude that:
- **Statement A**: Incorrect
- **Statement B**: Correct
- **Statement C**: Correct
- **Statement D**: Incorrect

Thus, the correct statements are **B and C**.