Question

A paramagnetic material has ${10^{28}}atoms/{m^3}$. Its magnetic susceptibility at temperature $350K$ is $2.8 \times {10^{ - 4}}$. Its susceptibility at $300K$ is:
(A) $2.672 \times {10^{ - 4}}$
(B) $3.726 \times {10^{ - 4}}$
(C) $3.267 \times {10^{ - 4}}$
(D) $3.672 \times {10^{ - 4}}$

Answer 1

Hint:
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
$\chi = \dfrac{c}{T}$, here $\chi $ is magnetic susceptibility of a paramagnetic material at the absolute temperature $T$ and $c$ 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
$\chi \propto \dfrac{1}{T}$
Removing the proportionality sign, we get
$\chi = \dfrac{c}{T}$ (1)
 where c is a constant for a given paramagnetic material and is of appropriate dimensions.
According to the question, at the temperature
${T_1} = 350K$${\chi _1} = 2.8 \times {10^{ - 4}}$
Substituting these in equation (1), we get
$2.8 \times {10^{ - 4}} = \dfrac{c}{{350}}$ (2)
According to the question, we have
${T_2} = 300K$
Let the magnetic susceptibility at this temperature be ${\chi _2}$
From (1)
${\chi _2} = \dfrac{c}{{300}}$ (3)
Dividing (3) by (2) we get
$\dfrac{{{\chi _2}}}{{2.8 \times {{10}^{ - 4}}}} = \dfrac{c}{{300}} \times \dfrac{{350}}{c}$
$\dfrac{{{\chi _2}}}{{2.8 \times {{10}^{ - 4}}}} = \dfrac{{350}}{{300}}$
On simplifying the above expression, we get
${\chi _2} = \dfrac{{350}}{{300}} \times 2.8 \times {10^{ - 4}}$
Finally, we have
${\chi _2} = 3.267 \times {10^{ - 4}}$
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.