As shown in Figure 23.26, the d-d transition of [Ti(H 2 O) 6 ]³⁺ produces an absorption maximum at a wavelength of about 500 nm . a. What is the magnitude of ∆ for [Ti(H 2 O) 6 ]³⁺ in kJ/mol? b. How would the magnitude of ∆ change if the H 2 O ligands in [Ti(H 2 O) 6 ]]³⁺ were placed with NH 3 ligands?

All right, hello everyone. So this question says that the hypothetical complex A H 206 2 positive has ad D transition that produces an absorption maximum at a wavelength of about 550 nanometers. Part one is asking what is the value for or rather what is the value of delta four A H2O 62 positive and kilo joules per mole. And part two is asking how would the magnitude of delta change if the h2o ligands in A H2O 62 positive were replaced with no two negative? All right. So here recall first and foremost that delta represents the crystal field splitting energy and is the difference in energy between the higher level and lower level orbitals. And so first, let's go ahead and calculate the value for delta. Now, here we are given information about a photon. And so we have to recall the formula to find the energy of that photo. Here, delta E is equal to HC divided by lambda H is planks constant C is the speed of light and lambda is the wavelength in meters. So let's go ahead and substitute these values H which is planks constant is equal to 6.626 multiplied by 10 to the negative 34th power dual seconds C is the speed of light. That's 3.00 multiplied by 10 to the eighth meters per second. And all of this is divided by LAMBDA which is the wavelength provided in meters. And so the photon has a wavelength of 550 multiplied by 10 to the negative ninth meters. thats for the photo. So now evaluating this expression for the energy of the photon that's going to be equal to 3.614182 multiplied by 10th, the negative 19th power joules per photon. However, here we don't have the correct units just yet because our answer is supposed to be reported in the units of kilo jules per bolt. Because of this, we're going to use Avogadro's number to convert from photons tools. And we also have to go from jewels to kilo joules. So let me go ahead and rewrite the answer here for delta E. That's 3.614182 multiplied by 10th of the negative 19th power joules per photon. I am going to use Avo Gajos number as a conversion factor to cancel out photons. This means that I have Ava Garos number thats six point 022 multiplied by 10 to the 23rd power vs in the numerator of my conversion factor and one mole of photons in the denominator, this cancels out photons and leaves me with units of jules per m. So now to go from jewels to kilo jewels, I'm also going to go ahead and divide this quantity by 1000 since 1000 joules equals one kilojoule. And so now after evaluating this expression and also rounding my answer to two significant figures, the value for delta of the hypothetical complex is equal to 2.2 multiplied by 10 to the second power ketel joules per mole. And that would be our answer for part one, which I'm going to rewrite on the top of the screen here. So now lets go ahead and talk about part two. Part two is asking how would the magnitude of delta change if the H2O ligands of the hypothetical complex were replaced with no two negative? And so for this part, recall that the strength of the ligands themselves and the ability to increase the energy gap decrease in the following order, just scroll down here to open up some space. But the order is as follows. The ligand that causes the largest value for delta is cyanide or C and negative. This is followed by I know two negative, followed by en and then NH three followed by H2O followed by fluoride and then chloride bromide. And lastly iodide is the ligand that results in the smallest value for delta. Because of this, we can see that an 02 negative is stronger than water as a ligand and therefore has a greater ability to increase the value of delta. This means that replacing H2O with no two negative is going to increase the magnitude of delta because no two negative creates a stronger field ligand and there you have it. So just to go ahead and recap our answers. For part one, the value for delta was 2.2 multiplied by 10th, the second power kilo joules per mole. And as for part two, it will increase, referring to the value of delta. And so with that being said, thank you so very much for watching. And I hope you found this helpful.

Ch.23 - Transition Metals and Coordination Chemistry

Brown - Chemistry: The Central Science 15th Edition

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Ch.23 - Transition Metals and Coordination Chemistry

Problem 54

Chapter 23, Problem 54

As shown in Figure 23.26, the d-d transition of [Ti(H₂O)₆]³⁺ produces an absorption maximum at a wavelength of about 500 nm .

a. What is the magnitude of ∆ for [Ti(H₂O)₆]³⁺ in kJ/mol?

b. How would the magnitude of ∆ change if the H₂O ligands in [Ti(H₂O)₆]]³⁺ were placed with NH₃ ligands?

Verified step by step guidance

Step 1: Understand that the d-d transition refers to the energy difference (∆) between the split d-orbitals in a transition metal complex. This energy difference corresponds to the wavelength of light absorbed.

Step 2: Use the formula for energy of a photon: E = \( \frac{hc}{\lambda} \), where h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and \( \lambda \) is the wavelength in meters.

Step 3: Convert the given wavelength from nanometers to meters (500 nm = 500 x 10^-9 m) and substitute the values into the energy formula to find the energy in joules.

Step 4: Convert the energy from joules to kJ/mol by using Avogadro's number (6.022 x 10^23 mol^-1) to account for the number of molecules in a mole.

Step 5: For part b, consider the spectrochemical series, which ranks ligands by their ability to split d-orbitals. NH3 is a stronger field ligand than H2O, so replacing H2O with NH3 would increase the magnitude of ∆.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Crystal Field Theory

Crystal Field Theory (CFT) explains how the arrangement of ligands around a central metal ion affects its electronic structure and energy levels. In transition metal complexes, the d-orbitals split into different energy levels due to the electrostatic interactions between the metal ion and the surrounding ligands. This splitting is crucial for understanding the absorption of light, as electrons can be promoted between these split d-orbitals, leading to color and spectral properties.

Ligand Field Strength

Ligand field strength refers to the ability of a ligand to influence the energy of the d-orbitals in a metal complex. Strong field ligands, like NH₃, cause a larger splitting of the d-orbitals compared to weak field ligands, such as H₂O. This difference in splitting affects the magnitude of the crystal field splitting energy (∆), which can be calculated from the wavelength of light absorbed during electronic transitions.

Absorption Spectroscopy

Absorption spectroscopy is a technique used to measure the amount of light absorbed by a substance at different wavelengths. In the context of transition metal complexes, the wavelength at which maximum absorption occurs corresponds to the energy difference between the split d-orbitals. By applying the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength, one can determine the value of ∆ in kJ/mol.

As shown in Figure 23.26, the d-d transition of [Ti(H2O)6]³⁺ produces an absorption maximum at a wavelength of about 500 nm .a. What is the magnitude of ∆ for [Ti(H2O)6]³⁺ in kJ/mol?b. How would the magnitude of ∆ change if the H2O ligands in [Ti(H2O)6]]³⁺ were placed with NH3 ligands?

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Key Concepts

Crystal Field Theory

Ligand Field Strength

Absorption Spectroscopy