Write a short essay to highlight the most significant

Question

Part 1:

a.) The vibrational frequencies of F₂, Cl₂, Br₂, and I₂ are 917 cm^-1, 525 cm^-1, 325 cm^-1, and 208 cm^-1. Which molecule (F₂, Cl₂, Br₂, or I₂) has the largest number of thermally accessible states? Make a prediction and explain your reasoning before you perform any calculations, but assume that all molecules are exposed to the same temperature.

b.) Treat the molecular vibrations as harmonic oscillators to calculate the vibrational partition function of the halogens and the population of their first three vibrational energy levels (i = 0,1,2) at 500 K.

	qv	p0	p1	p2
F2
Cl2
Br2
I2

c.) At what temperature would the vibrational partition function of fluorine be identical to the vibrational partition function of iodine calculated at 500 K?

Part 2:

Evaluate the rotational temperature Θ_R and the rotational partition function q_Rfor F₂, Cl₂, Br₂, and I₂ at 500 K. Is the high-temperature expression (q_R=T/(σ Θ_R) valid for all your calculations? The rotational constants for F₂, Cl₂, Br₂, and I₂ are 0.889 cm^-1, 0.244 cm^-1, 0.0821 cm^-1, and 0.0374 cm^-1.

	ΘR (K)	qR	High temperature expression valid?
F2
Cl2
Br2
I2

Part 3:

a.) Assume that all electronic ground states of the halogens are non-degenerate. Briefly explain why you can set the electronic partition function equal to one (q_E = 1) for all halogens at 500 K.

b.) Demonstrate that your simplification (q_E = 1) is justified by calculating the electronic partition function for bromine with the data provided below. Show your work!

Electronic energy level for Br2	Energy (cm-1)	Degeneracy
Ground state	0	1
First excited state	13905	6
Second excited state	15902	6
Third excited state	24000	2

Part 4:

Calculate the thermal de Broglie wavelength Λ and the translational partition function at 500 K and 1 bar for F₂, Cl₂, Br₂, and I₂.

	Λ (m)	qt
F2
Cl2
Br2
I2

Part 5:

Combine the partition functions of all relevant molecular modes and compute G_m (500 k) - G_m (0) for the four halogens. Assume standard pressure (1 bar).

	Total molecular partition function
F2
Cl2
Br2
I2

Part 6:

a.) Write a short essay to highlight the most significant trends in the data that you calculated in this assignment. Compare your data by chemical species and also by energy mode.

b.) Energy can be expressed in various units including J or cm^-1 among many other energy units. Convert the value of kT for T= 500 K into units of cm^-1. Sometimes the term kT is referred to as a scale of energy. Compare the value of kT at 500K to the energy values of the different modes for the halogen molecules. Briefly elaborate why kT can be a very useful energy scale in thermodynamics or spectroscopy.