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Question: The bond dissociation energies of H-H, C-C and C-H bonds respectively are 104.2, 83.1 and 98.9 kcal/...

The bond dissociation energies of H-H, C-C and C-H bonds respectively are 104.2, 83.1 and 98.9 kcal/mol. The electronegativity of carbon is:
A. 2.53
B. 2.51
C. 2.57
D. 2.52

Explanation

Solution

We can solve this question by keeping the following formula in mind that relates the electronegativity of the elements to the bond energies between them:
XC−XH=(eV)1/2E(C−H)−EC−C+EH−H2{{X}_{C}}-{{X}_{H}}={{(eV)}^{1/2}}\sqrt{{{E}_{(C-H)}}-\dfrac{{{E}_{C-C}}+{{E}_{H-H}}}{2}}
where, XC{{X}_{C}}is the electronegativity of carbon, XH{{X}_{H}} is the electronegativity of hydrogen, eV denotes electron volt per molecule and EH−H{{E}_{H-H}}, EC−C{{E}_{C-C}} and EC−H{{E}_{C-H}} are the bond energies of hydrogen-hydrogen, carbon-carbon, and carbon-hydrogen bonds respectively.

Complete Solution :
The bond dissociation energies of the following bonds are given as follows:
EH−H=104.2{{E}_{H-H}}=104.2kcal/mol
EC−C=83.1{{E}_{C-C}}=83.1kcal/mol
EC−H=98.8{{E}_{C-H}}=98.8kcal/mol
The difference in electronegativity of carbon and hydrogen can be given by the following equation:
XC−XH=(eV)1/2E(C−H)−EC−C+EH−H2{{X}_{C}}-{{X}_{H}}={{(eV)}^{1/2}}\sqrt{{{E}_{(C-H)}}-\dfrac{{{E}_{C-C}}+{{E}_{H-H}}}{2}}​​​
The standard value of hydrogen electronegativity, XH{{X}_{H}}​= 2.1 , and
eV = 0.04326 per molecule.

Substituting these values in the given equation, we get:

& {{X}_{C}}-2.1=(0.208)\sqrt{98.8-\dfrac{104.2+83.1}{2}} \\\ & \Rightarrow {{X}_{C}}-2.1=(0.208)(2.2694) \\\ & \Rightarrow {{X}_{C}}=2.572 \\\ \end{aligned}$$ **So, the correct answer is “Option C”.** **Note:** Bond dissociation energy is defined as “the amount of energy required to break one mole of bond of a particular type between the atoms in the gaseous state under standard conditions”. \- Electronegativity is defined as a measure of the tendency of an atom to attract a bonding pair of electrons. The Pauling scale is the most commonly used to measure electronegativity of the elements. \- Linus Pauling related the electronegativity difference between two atoms forming a bond to the bond energies of molecules. The larger the difference in the electronegativities of the bonded atoms, the greater is the bond dissociation energy.