This is a matching-type question where we need to associate the laws/processes in Column I with the corresponding physical phenomena in Column II. Let's analyze each item in Column I one by one.

Step 1: Analyze Column I (A) - Transition between two atomic energy levels

• A transition of an electron between two discrete energy levels within an atom results in the emission or absorption of a photon. The energy of the photon is equal to the energy difference between the two levels, i.e., $E_{photon} = \Delta E = E_{initial} - E_{final}$.
• (P) Characteristic X-rays: These are produced when an electron from a higher energy level (e.g., L-shell) makes a transition to a lower energy level (e.g., K-shell) to fill a vacancy. This is a transition between two atomic energy levels.
• (R) Hydrogen spectrum: The spectral lines (like Lyman, Balmer series) in the hydrogen spectrum are produced when an electron in a hydrogen atom de-excites from a higher energy level ($n_2$) to a lower energy level ($n_1$), emitting a photon. This is also a transition between two atomic energy levels.
• Therefore, (A) matches with both (P) and (R).

Step 2: Analyze Column I (B) - Electron emission from a material

• This phrase describes processes where electrons are ejected from a substance.
• (Q) Photoelectric effect: This is the phenomenon where electrons are emitted from a material (usually a metal) when electromagnetic radiation, such as light, hits the material. This perfectly matches the description.
• (S) $\beta$-decay: This is a type of radioactive decay in which a beta particle (a fast, energetic electron or positron) is emitted from an atomic nucleus. Since the nucleus is part of the material, this is also a form of electron emission from a material.
• Other phenomena like thermionic emission also fit this description but are not listed.
• Therefore, (B) matches with both (Q) and (S).

Step 3: Analyze Column I (C) - Mosley's law

• Mosley's law is an empirical law concerning the characteristic X-rays emitted by atoms. It states that the square root of the frequency ($\nu$) of the emitted X-ray is approximately proportional to the atomic number ($Z$) of the element. The formula is given by $\sqrt{\nu} = a(Z - b)$, where 'a' and 'b' are constants that depend on the type of X-ray line (e.g., K$\alpha$, L$\alpha$).
• (P) Characteristic X-rays: Mosley's law is exclusively related to characteristic X-rays.
• Therefore, (C) matches only with (P).

Step 4: Analyze Column I (D) - Change of photon energy into kinetic energy of electrons

• This describes a process where a photon is absorbed and its energy is transferred to an electron, which then possesses kinetic energy.
• (Q) Photoelectric effect: This is the quintessential example of this process. According to Einstein's photoelectric equation, $K_{max} = h\nu - \phi$, the energy of the incident photon ($h\nu$) is converted into the work function ($\phi$) to free the electron and the maximum kinetic energy ($K_{max}$) of the emitted electron.
• Therefore, (D) matches with (Q).

Step 5: Consolidate the matches and select the correct option

Based on the analysis above, we have the following matches:

• A $\to$ (P), (R)
• B $\to$ (Q), (S)
• C $\to$ (P)
• D $\to$ (Q)

Now, let's compare these matches with the given options:

• A: A $\to$(P); B $\to$(Q); C $\to$(P); D $\to$ (Q) - Incorrect because it misses A$\to$R and B$\to$S.
• B: A $\to$(R); B $\to$(Q, S); C $\to$(P); D $\to$ (Q, P) - Incorrect because it misses A$\to$P and incorrectly includes D$\to$P.
• C: A $\to$(P, R); B $\to$(Q, S); C $\to$(P); D $\to$ (Q) - This option perfectly matches our derived set of correspondences.
• D: A $\to$(P, R); B $\to$(Q, S); C $\to$(Q); D $\to$ (P) - Incorrect because C$\to$Q and D$\to$P are wrong.

Thus, the correct option is C.