Correct Statements Regarding Atomic Orbitals and Energies

To determine which of the given statements are true, let us analyze each statement individually:

1. Statement A: If two orbitals have the same value of $(n + l)$, the orbital with lower value of $n$ will have lower energy.

   • According to the $(n + l)$ rule (Bohr-Bury rule):
     • The energy of an orbital increases with an increase in the value of $(n + l)$.
     • When two orbitals have the same value of $(n + l)$, the orbital with a lower value of $n$ (principal quantum number) is situated closer to the nucleus and possesses lower energy.
   • Therefore, Statement A is TRUE.

2. Statement B: Energies of the orbitals in the same subshell increase with increase in atomic number.

   • As the atomic number ($Z$) increases, the nuclear charge increases.
   • A higher nuclear charge results in a stronger electrostatic force of attraction between the nucleus and the electron, which lowers (makes more negative) the energy of the orbitals.
   • Thus, the energy of orbitals in a given subshell decreases (becomes more negative) as the atomic number increases.
   • Therefore, Statement B is FALSE.

3. Statement C: The size of $2p_x$ orbital is less than the size of $3p_x$ orbital.

   • The size of an atomic orbital is primarily determined by the principal quantum number ($n$).
   • As $n$ increases, the region where the electron is most likely to be found extends further from the nucleus, increasing the size of the orbital.
   • Since $n = 2$ for the $2p_x$ orbital and $n = 3$ for the $3p_x$ orbital, the size of $2p_x$ is smaller than that of $3p_x$.
   • Therefore, Statement C is TRUE.

4. Statement D: Among $5f, 6s, 4d, 5p$ and $5d$ orbitals, none of the orbitals have 2 radial nodes.

   • The number of radial nodes for an orbital is given by the formula: $\text{Radial nodes} = n - l - 1$
   • Calculating the number of radial nodes for each orbital:
     • For $5f$: $n = 5, l = 3 \implies \text{Radial nodes} = 5 - 3 - 1 = 1$
     • For $6s$: $n = 6, l = 0 \implies \text{Radial nodes} = 6 - 0 - 1 = 5$
     • For $4d$: $n = 4, l = 2 \implies \text{Radial nodes} = 4 - 2 - 1 = 1$
     • For $5p$: $n = 5, l = 1 \implies \text{Radial nodes} = 5 - 1 - 1 = 3$
     • For $5d$: $n = 5, l = 2 \implies \text{Radial nodes} = 5 - 2 - 1 = 2$
   • Since the $5d$ orbital has 2 radial nodes, the claim that "none of the orbitals have 2 radial nodes" is incorrect.
   • Therefore, Statement D is FALSE.

Conclusion: Only statements A and C are true.

This corresponds to the option A and C only.