*"I have seen a medicine, that's able to breathe life into stone . . ."*.

In 2011, Russian researchers investigated a compound that appears to have tasted the medicine that Lafew describes in Shakespeare's play *All's Well That Ends Well*.

Consider a compound **B** of d-metal **A** whose chemical formula is \(CsACl_{n}\). When 0.948356g of this substance reacts with 0.23787g of \(CsF\) and an excess of fluorine gas, it is observed that 1.388106416g of a red solid **C** is formed and 0.3334g of chlorine gas is evolved.

**A** forms a chelate complex with two moles of an organic compound **D** which is a symmetric diketone whose formula is \(C_{5}H_{2}O_{2}F_{6}\). This complex is dissolved in toluene and mixed with another organic compound **W** to form a large solvate complex **Q**. The process of producing **W** from precursor \(\mu\) (shown below) proved to be inefficient.

To overcome this, the following sequence was introduced. We start with a compound *T* that is an acetal whose H-NMR contains 4 distinct signals and its intensities are in the ratio 1:1:4:6.

**W** is radical species of the formula \(C_{10}H_{15}N_{4}O_{2}\).

The final product mix contains regio-isomers that can be easily separated. We distinguish the isomers using a H-NMR machine that can detect coupling no greater than 3J values. **W** observes coupling at regions greater than 7.9 ppm, while **W'** observes coupling in the 5.9 - 7.2 ppm range.

The solvate complex forms many different crystals, one of these crystals contain a long chain of repeating units and we call this unit **G**. **G** can be considered to be an octahedral complex and at a transition temperature of 150K it changes from a trans form to a pseudo-cis form in a reversible manner without disturbing the integrity of the chain.

Let *n* denote the number of pseudo-cis units, it is given by the following statistical equation :

\(n = 10^{70}exp(\frac{-4637}{150-T})\) for all temperatures **T** less than 150K

When many units **G** morph into the pseudo-cis configuration the crystal shrinks in volume. The ratio of volume at **T** and the volume at 150K be governed by the empirical formula :

\(\frac{V_{T}}{V_{150}} = (1-\frac{n}{10^{5}})\)

We now define certain quantities:

\(\alpha\) is the number of rings in **W**

\(\beta\) is the valency of **A** in compound **C**

\(\gamma\) is the ratio between volumes at **T** = 120K (taken up to two decimal places)

Calculate the value of \(\alpha + \beta +100\gamma\)

**Bonus:**

- Deduce the formula of compounds
**A**-**C** - Deduce the structure of compounds
*T*-**W** - Deduce the structure of
**G**in trans configuration - If
**T**is varied periodically, what motion does the crystal mimic?

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