By Sunil Bhardwaj

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The vibrational energy of a harmonic oscillator is given by Schrodinger's relation. $${ E }_{ v } = \left( v + \frac { 1 }{ 2 } \right) h\omega \qquad \qquad \qquad ...(1)$$ where $$v$$ = vibrational quantum number having values 0, 1, 2, ..... etc.
$$h$$ = Planck's constant
$$\omega$$ = fundamental frequency of vibration $$\omega = \frac { 1 }{ 2\pi } \sqrt { \frac { K }{ \mu } }$$ Lets put this value in equation (1), we get, $${ E }_{ v } = \left( v + \frac { 1 }{ 2 } \right) \frac { h }{ 2\pi } \sqrt { \frac { K }{ \mu } }$$ For lowest energy level, $$v = 0$$ $${ E }_{ v } = \left( 0 + \frac { 1 }{ 2 } \right) \frac { h }{ 2\pi } \sqrt { \frac { K }{ \mu } }$$ $${ E }_{ v } = \frac { 1 }{ 2 } \frac { h }{ 2\pi } \sqrt { \frac { K }{ \mu } }$$ This energy possessed by molecule when $$v = 0$$ is called Zero Point Energy. The zero point energy indicates that the atoms can never be at rest, the molecules must vibrate even at $$0$$ K. i.e., at absolute zero, where translational and rotational energies are to be zero.

The vibrational frequency of HCl is $$2.988 \times { 10 }^{ 5 } { m }^{ -1 }$$. Calculate the zero point energy of the molecule.

Solution: We have wave number $$\overline { \upsilon } = 2.988 \times { 10 }^{ 5 } { m }^{ -1 }$$
Zero Point Energy is the vibrational energy at $$v = 0$$ $$\therefore { E }_{ 0 } = \frac { 1 }{ 2 } h\omega = \frac { 1 }{ 2 } h\upsilon = \frac { 1 }{ 2 } h\overline { \upsilon } c$$ $$= \frac { 1 }{ 2 } \left( 6.626 \times { 10 }^{ -34 } Js \right) \left( 2.988 \times { 10 }^{ 5 } { m }^{ -1 } \right) \left( 3 \times { 10 }^{ 8 } m/s \right)$$ $$\qquad = \frac { 1 }{ 2 } \left( 5.94 \times { 10 }^{ -20 } \right)$$ $$\qquad = 2.97 \times { 10 }^{ -20 } J$$

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