Question

Choose the correct reciprocal susceptibility $(1/\chi)$ vs. temperature (T) plot for paramagnetic substances above the critical temperature : 

 

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For $1/\chi$ vs $T$ plots: Ideal Paramagnet = passes exactly through the origin ($T=0$). Ferromagnet above $T_c$ = positive intercept on T-axis ($T = T_c$). Antiferromagnet above Neel temp = intersects the negative T-axis (extrapolates to $T = -\theta$).

Answer 1

The Correct Option is D

Step 1: Concept:
The problem requires identifying the correct graphical relationship between inverse magnetic susceptibility ($1/\chi$) and absolute Temperature ($T$) for a material that possesses a critical temperature (a Curie temperature, $T_c$) and behaves paramagnetically above it.

Step 2: Key Formula or Approach:
The language "paramagnetic substances above the critical temperature" specifically describes a Ferromagnetic material in its disordered state at high temperatures ($T > T_c$).
In this regime, the susceptibility obeys the Curie-Weiss Law:
\[ \chi = \frac{C}{T - T_c} \]
where $C$ is the Curie constant and $T_c$ is the Curie critical temperature.
Taking the reciprocal of this equation yields:
\[ \frac{1}{\chi} = \frac{T - T_c}{C} = \frac{1}{C}T - \frac{T_c}{C} \]

Step 3: Step-by-step Explanation:

• The equation $\frac{1}{\chi} = \frac{1}{C}T - \frac{T_c}{C}$ is in the form of a linear equation $y = mx + c$.

• The y-axis is $1/\chi$ and the x-axis is $T$.

• The slope ($m = 1/C$) is positive, indicating a straight line rising to the right.

• To find the x-intercept, set $1/\chi = 0$. This gives $0 = \frac{T - T_c}{C}$, which means $T = T_c$.

• Because the material is ferromagnetic, its critical temperature $T_c$ is a positive value ($T_c > 0$). Therefore, the straight line must intersect the positive Temperature axis.

• Looking at the given plots:
- Plot (1) shows a curve.
- Plot (2) shows a constant value.
- Plot (3) shows a straight line passing through the origin. This represents the Curie Law ($\chi = C/T$) for an ideal paramagnet with no critical temperature interactions ($T_c = 0$).
- Plot (4) shows a straight line intersecting the positive T-axis at $T_c$. This perfectly matches our Curie-Weiss derivation.

Step 4: Final Answer:
Graph 4 correctly represents this relationship, matching option (D).