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Question: A paramagnetic material has \({10^{28}}atoms/{m^3}\). Its magnetic susceptibility at temperature \(3...

A paramagnetic material has 1028atoms/m3{10^{28}}atoms/{m^3}. Its magnetic susceptibility at temperature 350K350K is 2.8×10−42.8 \times {10^{ - 4}}. Its susceptibility at 300K300K is:
(A) 2.672×10−42.672 \times {10^{ - 4}}
(B) 3.726×10−43.726 \times {10^{ - 4}}
(C) 3.267×10−43.267 \times {10^{ - 4}}
(D) 3.672×10−43.672 \times {10^{ - 4}}

Explanation

Solution

To solve this question, we need to use the curie’s law for a paramagnetic material. And then using the information given, we can get the final value.
Formula used: The formula used in solving this question is
χ=cT\chi = \dfrac{c}{T}, here χ\chi is magnetic susceptibility of a paramagnetic material at the absolute temperature TT and cc is a constant.

Complete step by step answer:
We know by curie’s law, that if a paramagnetic material is heated, its magnetic susceptibility varies in an inverse proportion with the absolute temperature, that is
χ∝1T\chi \propto \dfrac{1}{T}
Removing the proportionality sign, we get
χ=cT\chi = \dfrac{c}{T} (1)
where c is a constant for a given paramagnetic material and is of appropriate dimensions.
According to the question, at the temperature
T1=350K{T_1} = 350K χ1=2.8×10−4{\chi _1} = 2.8 \times {10^{ - 4}}
Substituting these in equation (1), we get
2.8×10−4=c3502.8 \times {10^{ - 4}} = \dfrac{c}{{350}} (2)
According to the question, we have
T2=300K{T_2} = 300K
Let the magnetic susceptibility at this temperature be χ2{\chi _2}
From (1)
χ2=c300{\chi _2} = \dfrac{c}{{300}} (3)
Dividing (3) by (2) we get
χ22.8×10−4=c300×350c\dfrac{{{\chi _2}}}{{2.8 \times {{10}^{ - 4}}}} = \dfrac{c}{{300}} \times \dfrac{{350}}{c}
χ22.8×10−4=350300\dfrac{{{\chi _2}}}{{2.8 \times {{10}^{ - 4}}}} = \dfrac{{350}}{{300}}
On simplifying the above expression, we get
χ2=350300×2.8×10−4{\chi _2} = \dfrac{{350}}{{300}} \times 2.8 \times {10^{ - 4}}
Finally, we have
χ2=3.267×10−4{\chi _2} = 3.267 \times {10^{ - 4}}
This is the required magnetic susceptibility at the second temperature.
Hence, the correct answer is option (C).

Note:
Do not get confused by the value of the density of the atoms given in the question. It is just useless information given in the question, just to distract from the main approach to the solution. It is not related to any step of the main solution, as can be observed above. There is a second formula for the magnetic susceptibility too, which requires the use of this value. But that formula is not applicable here. So, just ignore this value and move ahead with your approach to solve the question without getting distracted.