Calculate the energy difference in cm^-1 and kJ mol^-1 between the J=0 and J=1 rotational levels

Calculate the energy difference in cm^-1 and kJ mol^-1...

Key Concepts

Energy Unit Conversion

Energy differences in molecular spectroscopy are often given in wavenumbers (cm?¹), but for practical chemical and thermodynamic calculations, it is useful to convert these values into energy per mole (kJ/mol). This involves using the appropriate physical constants such as Planck’s constant, the speed of light, and Avogadro’s number to bridge the gap between microscopic energy transitions and macroscopic energy units.

Reduced Mass

The reduced mass is a key parameter in the study of diatomic molecules, calculated using ? = (m1*m2)/(m1+m2), where m1 and m2 are the masses of the two atoms. This concept is important because it influences the moment of inertia and therefore the spacing of the rotational energy levels, as isotopic substitution (such as replacing hydrogen with deuterium) alters the reduced mass, changing the energy differences between quantum rotational states.

Rotational Energy Levels

In quantum mechanics, the rotational energy of a diatomic molecule is quantized and is given by the expression E(J) = B J (J+1), where B is the rotational constant and J is the rotational quantum number. This concept is fundamental to understanding how energy differences between rotational states (for example, between J = 0 and J = 1) are determined, as the spacing between these energy levels is a direct result of the quantized angular momentum of the molecule.

Moment of Inertia

The moment of inertia of a diatomic molecule plays a crucial role in its rotational energy levels. It is defined as I = ?r², where ? is the reduced mass of the molecule and r is the internuclear distance. A change in either the reduced mass or internuclear distance will affect the moment of inertia, and hence the rotational constant B, which ultimately impacts the energy difference between rotational states.

Transcript

00:01 Hi there, so for this problem we need to compute the difference in the rotational energy for the rotational energy of this molecule minus the rotational energy for this other molecule.

00:33 Now, in this case, we need first to calculate the mass, the reduced mass because we know that the rotational energy is defined as each part is squared divided by two times the moment of inertia and in this case we know that the moment of inertia is defined as the rest energy times the distance square so we can substitute that in here so we will have that the rotational energy is equal to h bar square divided by two times the reduced mass times the distance in the equilibrium square.

01:38 So for this we need first to calculate the reduced mass.

01:52 So in this case for this first molecule we will have that the reduced mass is equal to 39 .102 units.

02:15 This is for potassium.

02:20 And this times the mass of chlorine, in this case, is 34.

02:28 34 .969 units.

02:33 And this divided by the sum of 3 .3.

02:35 These two values.

02:42 So from this we obtain a reduced mass of 18 .46 units.

02:51 So that's the reduced mass.

02:55 And now for the other molecule, we will have that the reduced mass is equal.

03:07 In this case, the mass for potassium is the same as before, 39 .102 units.

03:15 But the mass for chlorine changes, so that is 34 .966 units, and this divided by the sum of these two values.

03:39 So from this, we obtain a reduced mass for this molecule of 19 units.

03:46 And we also know that the distance of equilibrium in potassium chlorine is equal to 0 .267 nanometers.

04:04 So with this, we can just find the energy for each of this and substitute the values that we obtain and then obtain the energy.

04:16 So we start with the first one.

04:20 So we will find that the rotational energy for potassium 35 is equal to...

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