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 [tex]300 K[/tex], where [tex]n=4[/tex].
Answer (2)
The magnetic susceptibility for one kg mole of Cr^{2+} ions at 300 K is approximately 100, calculated using principles of magnetism involving unpaired electrons and thermodynamic equations. It is derived from the magnetic moment and Curie's law. This result helps in understanding the magnetic properties of materials containing these ions.
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Calculate the magnetic moment using μ = n ( n + 2 ) μ B , where n = 4 and μ B = 9.274 × 1 0 − 24 J / T , resulting in μ ≈ 4.543 × 1 0 − 23 J / T .
Calculate the magnetic susceptibility using Curie's law: χ m = 3 k B T N A μ 2 , where N A = 6.022 × 1 0 23 m o l − 1 , k B = 1.381 × 1 0 − 23 J / K , and T = 300 K , resulting in χ m ≈ 0.100 .
Multiply the result by 1000 to account for 1 kg mole: χ t o t a l = 1000 × χ m .
The magnetic susceptibility for one kg mole of C r 2 + ions at 300 K is 100 .
Explanation
Problem Analysis We are given the electronic configuration of C r 2 + ion as 3 d 4 . We are also given that the number of unpaired electrons n = 4 . The temperature T = 300 K . We need to calculate the magnetic susceptibility for a salt containing one kg mole of C r 2 + ions.
Calculating Magnetic Moment First, we need to calculate the magnetic moment μ using the formula μ = n ( n + 2 ) μ B where n is the number of unpaired electrons and μ B is the Bohr magneton, which is equal to 9.274 × 1 0 − 24 J / T .
Magnetic Moment Calculation Substituting n = 4 into the formula, we get μ = 4 ( 4 + 2 ) × 9.274 × 1 0 − 24 = 24 × 9.274 × 1 0 − 24 ≈ 4.899 × 9.274 × 1 0 − 24 ≈ 4.543 × 1 0 − 23 J / T
Calculating Magnetic Susceptibility Next, we calculate the magnetic susceptibility χ m using Curie's law: χ m = 3 k B T N A μ 2 where N A is Avogadro's number ( 6.022 × 1 0 23 m o l − 1 ), k B is Boltzmann's constant ( 1.381 × 1 0 − 23 J / K ), and T is the temperature in Kelvin.
Magnetic Susceptibility Calculation Substituting the values, we get χ m = 3 × 1.381 × 1 0 − 23 × 300 6.022 × 1 0 23 × ( 4.543 × 1 0 − 23 ) 2 = 124.29 × 1 0 − 23 6.022 × 1 0 23 × 2.064 × 1 0 − 45 = 124.29 × 1 0 − 23 12.438 × 1 0 − 22 ≈ 0.100
Total Magnetic Susceptibility Since we have 1 kg mole, we need to multiply the result by 1000: χ t o t a l = 1000 × χ m = 1000 × 0.100 = 100
Final Answer Therefore, the magnetic susceptibility for a salt containing one kg mole of C r 2 + ions at 300 K is approximately 100.
Examples
Magnetic susceptibility is a fundamental property of materials that describes how much a material will become magnetized in an applied magnetic field. This property is crucial in designing various electronic devices, such as transformers, inductors, and magnetic sensors. For instance, in medical imaging, contrast agents containing paramagnetic ions like C r 2 + enhance the visibility of specific tissues by altering their magnetic properties, which is directly related to the magnetic susceptibility. Understanding and calculating magnetic susceptibility helps engineers and scientists select appropriate materials for specific applications and optimize the performance of magnetic devices.
