Matching Thermodynamic and Physical Processes with Energy Changes

To determine the correct matches, let us analyze each process described in Column II and evaluate it against the statements in Column I:

1. Process (p): A capacitor, initially uncharged, is connected to a battery.

   • Initially, the stored electrostatic potential energy of the capacitor is zero ($U_i = 0$).
   • When connected to a battery of potential $V$, the capacitor charges and stores electrostatic energy given by:
     $U_f = \frac{1}{2}CV^2 > 0$
   • Thus, the energy of the system increases.
   • Therefore, (p) matches with (A).

2. Process (q): A gas in an adiabatic container fitted with an adiabatic piston is compressed by pushing the piston.

   • For an adiabatic process, heat exchange with the surroundings is zero ($Q = 0$).
   • Work is done on the gas by an external agent by pushing the piston, meaning mechanical energy is supplied to the gas:
     $W_{\text{by gas}} < 0 \implies \Delta U = -W_{\text{by gas}} > 0$
   • The mechanical work done on the system increases the internal energy of the gas, which manifests as an increase in the temperature, i.e., the average kinetic energy of the random motion of its molecules.
   • Hence, the total energy of the system increases, and mechanical energy is converted into the energy of random thermal motion of its constituent parts.
   • Therefore, (q) matches with (A) and (B).

3. Process (r): A gas in a rigid container gets cooled due to the colder atmosphere surrounding it.

   • Since the container is rigid, the volume is constant, so no work is done ($W = 0$).
   • Heat flows out of the gas to the colder surroundings ($Q < 0$), which causes the internal energy to decrease:
     $\Delta U = Q < 0$
   • Hence, the energy of the system decreases.

4. Process (s): A heavy nucleus, initially at rest, fissions into two fragments of nearly equal masses and some neutrons are emitted.

   • In a nuclear fission reaction, the sum of the rest masses of the fission fragments and emitted neutrons is strictly less than the rest mass of the parent nucleus due to the mass defect $\Delta m$:
     $\Delta m = m_{\text{parent}} - \sum m_{\text{products}} > 0$
     Thus, the mass of the system decreases.
   • By Einstein's mass-energy equivalence, this decrease in mass (which is part of the system's rest internal energy) is converted into the kinetic energy of the resulting fragments and emitted neutrons:
     $Q_{\text{value}} = (\Delta m)c^2 = K_{\text{fragments}} + K_{\text{neutrons}}$
   • The kinetic energy of the moving fragments constitutes macroscopic/mechanical energy of the particles. Thus, internal energy is converted into mechanical energy.
   • Therefore, (s) matches with (C) and (D).

5. Process (t): A resistive wire loop is placed in a time-varying magnetic field perpendicular to its plane.

   • By Faraday's Law of Electromagnetic Induction, a time-varying magnetic flux induces an electromotive force (emf) $\mathcal{E} = -\frac{d\Phi_B}{dt}$ and an electric current in the resistive wire.
   • The induced electric field does work on the charge carriers, dissipating energy via Joule heating ($P = I^2 R$), which raises the temperature and thermal/internal energy of the loop.
   • Thus, the energy of the system increases.
   • Therefore, (t) matches with (A).

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Summary of Matches:

• (A) $\to$ p, q, t
• (B) $\to$ q
• (C) $\to$ s
• (D) $\to$ s