2. The electronic configuration of a [tex]Cr ^{2+}[/tex] ion is [tex]$3 d^4 4 s^0$[/tex]. Calculate the magnetic susceptibility for a salt containing one kg mole of [tex]Cr ^{2+}[/tex] ions at 300 K.
[tex] [/tex]
[tex]$(1.25 \times 10^{-4})$[/tex]
(Hint : Orbital angular momentum of [tex]Cr ^{2+}[/tex] is quenched by the presence of the crystal field).
Answer (2)
To calculate the magnetic susceptibility for a salt containing one kg mole of Cr^{2+} ions at 300 K, we find the effective magnetic moment to be approximately 4.543 x 10^{-23} J/T. The magnetic susceptibility is calculated using the number of ions, effective moment, Boltzmann's constant, and temperature, resulting in a value of approximately 100.08. Thus, χ m for Cr^{2+} ions in this scenario is approximately 100.08.
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Calculate the effective magnetic moment using the formula μ e ff = g S ( S + 1 ) μ B , where S = 2 and g = 2 , resulting in μ e ff ≈ 4.543 × 1 0 − 23 J / T .
Determine the number of C r 2 + ions: N = 1000 × 6.022 × 1 0 23 = 6.022 × 1 0 26 .
Use the formula for magnetic susceptibility: χ m = 3 k T N μ e ff 2 .
Calculate the magnetic susceptibility: χ m ≈ 100.08 .
100.08
Explanation
Problem Analysis The problem asks us to calculate the magnetic susceptibility of a salt containing C r 2 + ions. We are given that the electronic configuration of C r 2 + is 3 d 4 4 s 0 , the salt contains 1 kg-mole of C r 2 + ions, and the temperature is 300 K. We are also given a hint that the orbital angular momentum is quenched by the crystal field. This means we only need to consider the spin contribution to the magnetic moment.
Formula for Magnetic Susceptibility We will use the formula for magnetic susceptibility: χ m = 3 k T N μ 2 where:
N is the number of ions
μ is the magnetic moment
k is the Boltzmann constant
T is the temperature
Calculating Effective Magnetic Moment First, we need to calculate the effective magnetic moment μ e ff . Since the orbital angular momentum is quenched, we use the spin-only formula: μ e ff = g S ( S + 1 ) μ B where:
g is the g-factor (approximately 2 for spin-only contribution)
S is the total spin quantum number
μ B is the Bohr magneton
Determining Spin Quantum Number and Calculating Magnetic Moment For C r 2 + with electronic configuration 3 d 4 , there are 4 unpaired electrons. Therefore, the total spin quantum number is: S = 2 n = 2 4 = 2 Since the orbital contribution is quenched, g = 2 . The Bohr magneton μ B = 9.274 × 1 0 − 24 J / T . Therefore, μ e ff = 2 2 ( 2 + 1 ) μ B = 2 6 μ B = 2 6 × 9.274 × 1 0 − 24 J / T
Calculating Magnetic Moment Value μ e ff = 2 6 × 9.274 × 1 0 − 24 J / T ≈ 4.543 × 1 0 − 23 J / T
Calculating Number of Ions Next, we calculate the number of C r 2 + ions, N . We have 1 kg-mole, which is 1000 moles. Avogadro's number is N A = 6.022 × 1 0 23 ions/mole. Therefore, N = 1000 × N A = 1000 × 6.022 × 1 0 23 = 6.022 × 1 0 26
Calculating Magnetic Susceptibility The Boltzmann constant is k = 1.38 × 1 0 − 23 J / K , and the temperature is T = 300 K . Now we can calculate the magnetic susceptibility: χ m = 3 k T N μ e ff 2 = 3 × ( 1.38 × 1 0 − 23 ) × 300 ( 6.022 × 1 0 26 ) × ( 4.543 × 1 0 − 23 ) 2
Final Calculation χ m = 1.242 × 1 0 − 20 ( 6.022 × 1 0 26 ) × ( 2.064 × 1 0 − 45 ) = 1.242 × 1 0 − 20 1.243 × 1 0 − 18 ≈ 100.08
Final Answer Therefore, the magnetic susceptibility for the salt is approximately 100.08.
Examples
Magnetic susceptibility is a fundamental property of materials that describes how much a material will become magnetized in an applied magnetic field. Understanding magnetic susceptibility is crucial in various applications, such as designing MRI contrast agents, developing new magnetic storage devices, and characterizing the magnetic behavior of minerals in geology. For instance, in environmental science, measuring the magnetic susceptibility of soil samples can help identify pollution sources, as certain pollutants alter the magnetic properties of the soil. In medicine, contrast agents with high magnetic susceptibility enhance the visibility of specific tissues or organs during MRI scans, aiding in diagnosis.
