Question

The magnetic susceptibility of magnesium at $300\text{ K = 1}\text{.2 }\times \text{1}{{\text{0}}^{5}}$. At what temperature will it’s magnetic susceptibility become $1.44\times {{10}^{5}}$?

Answer 1

Hint As susceptibility is inversely proportional to temperature, hence on comparison of two equations, $\text{T}$ can be found.
Formula used $\text{susceptibility}\propto \dfrac{1}{\text{temperature}}$, where $\propto$ is the sign of proportionality.

Complete Step by step solution
It is given that,
$\text{X}{{\text{m}}_{1}}=1.25\times {{10}^{5}}$ at temperature ${{\text{T}}_{1}}=300\text{ K}$
$\text{X}{{\text{m}}_{2}}=1.44\times {{10}^{5}}$
We have to calculate temperature ${{\text{T}}_{2}}$.
We know that susceptibility is inversely proportional to temperature.
i.e. $\text{X}{{\text{m}}_{1}}\propto \dfrac{1}{{{\text{T}}_{1}}}$ …..(1)
$\text{X}{{\text{m}}_{2}}=\dfrac{1}{{{\text{T}}_{2}}}$ …..(2)
Combining equations (1) and (2), we get

& \dfrac{\text{X}{{\text{m}}_{1}}}{\text{X}{{\text{m}}_{2}}}=\dfrac{{{\text{T}}_{2}}}{{{\text{T}}_{1}}} \\\ & {{\text{T}}_{2}}={{\text{T}}_{1}}.\dfrac{\text{X}{{\text{m}}_{1}}}{\text{X}{{\text{m}}_{2}}} \\\ & {{\text{T}}_{2}}=300\times \dfrac{1.25\times {{10}^{5}}}{1.44\times {{10}^{5}}} \\\ & {{\text{T}}_{2}}=260.4\text{K} \end{aligned}$$ Hence, susceptibility of magnesium becomes $$1.44\times {{10}^{5}}$$ at temperature, $${{\text{T}}_{2}}=260.4\text{ K}$$. **Note:** Magnetic susceptibility is the measure of how much a material gets magnetized in an applied magnetized in an applied magnetic field. Paramagnetic susceptibility (as in case of magnesium) decreases with temperature because high temperature causes greater thermal vibration of atoms, which interferes with alignment of magnetic dipoles.