Amongst the following compounds
(I) `H_(5)P_(3)O_(10)`
(II) `H_(6)P_(4)O_(13)`
(III) `H_(5)P_(5)O_(15)`
(IV) `H_(7)P_(5)O_(16)`
non-cyclic phosphates are:

Options

A. I,III
B. I,II,III
C. I,II,IV
D. I,II,III,IV

Answer

To determine which of the given compounds are non-cyclic phosphates, we need to analyze the molecular formulas provided:

1. **Identify the compounds:**
- (I) \( H_5P_3O_{10} \)
- (II) \( H_6P_4O_{13} \)
- (III) \( H_5P_5O_{15} \)
- (IV) \( H_7P_5O_{16} \)

2. **Understand the concept of cyclic and non-cyclic phosphates:**
- Cyclic phosphates are typically formed from metaphosphoric acid units, which have the formula \( HPO_3 \).
- For a compound to be cyclic, the number of phosphorus atoms (P) must be a multiple of the number of oxygen atoms (O) in the repeating unit. For example, \( H_nP_nO_{3n} \) indicates a cyclic structure.

3. **Check each compound for cyclic structure:**
- For a compound to be cyclic, the ratio of P and O should follow the pattern of \( H_nP_nO_{3n} \).

- **Compound (I) \( H_5P_3O_{10} \)**:
- Here, \( n = 3 \) gives \( O = 3n = 9 \) (not matching 10). So, it is non-cyclic.

- **Compound (II) \( H_6P_4O_{13} \)**:
- Here, \( n = 4 \) gives \( O = 3n = 12 \) (not matching 13). So, it is non-cyclic.

- **Compound (III) \( H_5P_5O_{15} \)**:
- Here, \( n = 5 \) gives \( O = 3n = 15 \) (matches). So, it is cyclic.

- **Compound (IV) \( H_7P_5O_{16} \)**:
- Here, \( n = 5 \) gives \( O = 3n = 15 \) (not matching 16). So, it is non-cyclic.

4. **Conclusion:**
- The non-cyclic phosphates are (I) \( H_5P_3O_{10} \), (II) \( H_6P_4O_{13} \), and (IV) \( H_7P_5O_{16} \).
- The cyclic phosphate is (III) \( H_5P_5O_{15} \).

Thus, the final answer is that the non-cyclic phosphates are:
- (I) \( H_5P_3O_{10} \)
- (II) \( H_6P_4O_{13} \)
- (IV) \( H_7P_5O_{16} \)