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I understand the equation to use, and I know the answer but I do not understand what to do with...

Question:

I understand the equation to use, and I know the answer but I do not understand what to do with the rings and the pi bonds to get the correct answer

compound x has 50 carbons, 3 pi bonds, and 3 rings, how many hydrogens does x have?

Index of Hydrogen Deficiency:

The number of rings and {eq}\pi {/eq} bonds in a structure can be determined from the number of H atoms in the structure which is quite useful for analysis of organic compounds. This is because it provides the first information on how the atoms in the structural formula are connected to each other. The number of rings and {eq}\pi {/eq} bonds individually cannot be defeated, however, it provides a reference point for the interpretation of other data such as NMR and IR data. The index of hydrogen deficiency is also useful because it provides a simple method to calculate the number of H atoms in a structure without needing to count each individual H atom.

Answer and Explanation: 1

The general formula for an acyclic alkane is {eq}C_nH_{2n + 2} {/eq}. If the molecule contains a ring then the number of H atoms in a structure must decrease by two so that the C atoms can make a ring. Therefore a cyclic alkane would have a general formula of {eq}C_nH_{2n + 2 - 2x} {/eq} where x is the number of rings. If a molecule instead has a {eq}\pi {/eq} bond that means there must be one less H atom on each C participating in the {eq}\pi {/eq} bond. Therefore the general formula of an alkene would be {eq}C_nH_{2n + 2 - 2y} {/eq} where y is the number of {eq}\pi {/eq} bonds. It can be seen that the effect of rings and {eq}\pi {/eq} bonds is to decrease the number of H atoms by the same amount. The total number of rings and {eq}\pi {/eq} bonds in a structure can be determined by comparing the number of H atoms in the compound to the number of H atoms in the corresponding acyclic hydrocarbon with the same number of C atoms. This comparison between the number of hydrogens is called the index of hydrogen deficiency, or IDH. The IDH is the total number of rings and {eq}\pi {/eq} bonds which calculated by

{eq}IDH = \frac 12(\text {# of H in corresponding alkane - # of H in compound}) {/eq}

If a compound contained 50 C atoms then the corresponding acyclic alkane would be {eq}C_{50}H_{102} {/eq}. Since the compound has three {eq}\pi {/eq} bonds and three rings that means the IDH is

{eq}IDH = 3 + 3 \\ IDH = 6 {/eq} The number of H atoms in the structure can now be calculated

{eq}IDH = \frac 12(\text {# of H in corresponding alkane - # of H in compound} ) \\ 6 = \frac 12(102 - \text {# of H in compound}) \\ \text {# of H in compound} = 102 - 6 \times 2 \\ \text {# of H in compound} = 90 {/eq}

Therefore the number of H atoms in the compound is 90.


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