The reduced mass of a diatomic molecule is defined as m* = m_1m_2/m_1 + m_2 wher

Question

The reduced mass of a diatomic molecule is defined as m* = m_1m_2/m_1 + m_2 where m1 and m2 are the respective mass of each atom. The moment of inertia of a diatomic molecule is defined as I = m*R^2 where R is the bond length of the molecule. Estimate the molar heat capacity C_v of each molecule at 6 K assuming they behave as ideal gases. Provide quantitative justification to your answer. Using the given values of force constant k and bond length f?, determine the energy required to excite the first vibrational and the first rotational level for each molecule. Pay close attention to the units and make sure that they are consistent with SI units. Estimate the molar heat capacity C_v of each molecule at room temperature assuming they behave as ideal gases. Provide quantitative justification to your answer. Estimate the molar heat capacity C_v of each molecule at 4000 K assuming they behave as id3al gases. Provide quantitative justification to your answer.

Explanation / Answer

1. For vibration, Ev = ( v + 0.5) . (1/2c) . (k/m*) m-1

To excite first vibrational state to second vibrational state, dEv

= ( 2 + 0.5) . (1/2c) . (k/ m*) - ( 1 + 0.5) . (1/2c) . (k/ m*)

= (1/2c) . (k/ m*)

For NO: m* = 0.0075 kg

So, dEv = (1/2c) . (k/ m*)

= 2.39 * 10-7 m-1

For CO : m* = 0.0068 kg

So, dEv = (1/2c) . (k/ m*)

= 2.76 * 10-7 m-1

For HI : m* = 0.00099 kg

So, dEv = (1/2c) . (k/ m*)

= 3.01 * 10-7 m-1

For HBr :

m* = 0.00098 kg

So, dEv = (1/2c) . (k/ m*)

= 3.42 * 10-7 m-1

For rotation : dvJ = 2. ( h/82Ic) (J+1)

For NO : I= m*r2 = 1.1 * 10-22 kg m2

dvJ = 2. ( h/82Ic) (J+1)....[ J=1]

= 1.018 * 10-21 m-1

For CO : I= m*r2 = 0.97 * 10-22 kg m2

dvJ = 2. ( h/82Ic) (J+1)

= 1.14 * 10-21 m-1

For HI : I= m*r2 = 0.26 * 10-22 kg m2

dvJ = 2. ( h/82Ic) (J+1)

= 4.36 * 10-21 m-1

For HBr: I= m*r2 = 0.19 * 10-22 kg m2

dvJ = 2. ( h/82Ic) (J+1)

= 5.7 * 10-21 m-1

2,3,4. For ideal gases molar heat capacity does not depend on temperature.

For an ideal diatomic gas ,

Cv

= dQv/ dT

= d( 3/2 RT + RT + RT)/dT

= 7/2 R

= 29.1 J

Thus for all diatomic gases the value of Cv will be 29.1 J.

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