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Phase Transitions Questions and Answers

1.

According to Weiss molecular theory near the critical point the isothermal susceptibility of the magnetic system behaves as :

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: B

Near the critical point \(\chi_{T} \propto |T - T_{C}|^{-1}\).

In Weiss's molecular field theory each spin feels the applied field plus an internal field proportional to the magnetisation. Solving self-consistently above \(T_{C}\) gives the Curie-Weiss law:

\[ \chi = \frac{C}{T - T_{C}} \]

So the susceptibility diverges with critical exponent \(\gamma = 1\), the mean-field value. The exponent 1/2 belongs to the magnetisation below \(T_{C}\) (\(M \propto (T_{C} - T)^{1/2}\), \(\beta = 1/2\)), not the susceptibility, and a positive power would make \(\chi\) vanish at the critical point instead of diverging.

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2.

Second sound in superfluid helium is :

Exam Year(s)

MH SET - 2025, SET Exam - 2025,

Answer: A

Second sound is the wave motion of entropy and temperature.

In the two-fluid model of He II, the liquid is a mixture of a superfluid component, which carries no entropy, and a normal component, which carries all of it. In ordinary (first) sound both components move together, giving a pressure and density wave. In second sound they move in opposite directions with no net mass flow, so the total density and pressure stay almost constant while the local entropy, and with it the temperature, oscillates. It is a temperature wave, excited by a heater rather than a loudspeaker.

The pressure and density waves describe first sound, and the motion of the normal component alone is not second sound, which needs the counterflow of both components.

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