Detailed Solution:
According to Crystal Field Theory (CFT), the magnetic properties of a complex depend on whether electrons pair up or remain unpaired. This is determined by the Crystal Field Splitting Energy ($\Delta_0$) compared to the Pairing Energy (P). Strong field ligands (e.g., $CN^-, CO$) cause a large splitting ($\Delta_0 > P$), forcing electrons to pair up (low-spin, often diamagnetic). Weak field ligands (e.g., $Cl^-, F^-$) cause a small splitting ($\Delta_0 < P$), allowing electrons to occupy higher energy orbitals before pairing (high-spin, paramagnetic). Thus, the ligand field strength is the primary determining factor.

Detailed Solution:
Step 1: Identify the electronic configuration for a high-spin $d^4$ octahedral complex. High spin implies a weak ligand field where $\Delta_0 < P$. The electrons will occupy both $t_{2g}$ and $e_g$ sets before pairing.
Step 2: The configuration is $t_{2g}^3 \ e_g^1$.
Step 3: Apply the CFSE formula for octahedral complexes:
$CFSE = [(-0.4 \times n_{t_{2g}}) + (0.6 \times n_{e_g})] \Delta_0 + mP$
Where $n_{t_{2g}} = 3$ and $n_{e_g} = 1$. Since there is no forced pairing, $m = 0$.
$CFSE = [(-0.4 \times 3) + (0.6 \times 1)] \Delta_0$
$CFSE = [-1.2 + 0.6] \Delta_0$
$CFSE = -0.6 \Delta_0$

Detailed Solution:
According to the Second Law of Thermodynamics, a process is spontaneous if it leads to an increase in the total entropy of the universe.
The total entropy change is given by: $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$.
Therefore, the criterion for spontaneity is $\Delta S_{universe} > 0$, which mathematically equates to $\Delta S_{system} + \Delta S_{surroundings} > 0$.

Detailed Solution:
A Pourbaix diagram plots Potential (E) on the y-axis against pH on the x-axis.
The given half-reaction is $Fe^{2+} + 2e^- \rightleftharpoons Fe_{(s)}$. Notice that $H^+$ or $OH^-$ ions are not involved in this equilibrium.
Because the Nernst equation for this specific reaction does not contain a concentration term for $[H^+]$, the equilibrium potential $E$ depends only on the concentration of $Fe^{2+}$.
Therefore, on the Pourbaix diagram, this reaction is represented by a horizontal line, meaning its potential is completely pH independent.

Detailed Solution:
1. Salient Features of Crystal Field Theory (CFT):

• Electrostatic Interaction: The central metal ion and the surrounding ligands are treated as point charges (if anions) or point dipoles (if neutral molecules like $NH_3$ or $H_2O$). The interaction between the metal ion and the ligands is purely electrostatic in nature; no covalent bond formation (orbital mixing) is considered.
• Degeneracy of d-orbitals: In a free, isolated gaseous metal ion, all five d-orbitals ($d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, d_{z^2}$) possess the same energy (they are degenerate).
• Spherically Symmetrical Field: When ligands approach the central metal ion symmetrically from all directions, the energy of all five d-orbitals increases equally due to electrostatic repulsion between the ligand electrons and the metal's d-electrons.
• Crystal Field Splitting: Because ligands approach along specific geometries (e.g., octahedral, tetrahedral) rather than spherically, the d-orbitals that lie directly in the path of the approaching ligands experience greater electrostatic repulsion and thus higher energy than those lying between the axes.
  In an octahedral field, the ligands approach along the axes. Thus, the axial orbitals ($d_{x^2-y^2}, d_{z^2}$) form the higher energy $e_g$ set, while the non-axial orbitals ($d_{xy}, d_{yz}, d_{zx}$) form the lower energy $t_{2g}$ set.
• Crystal Field Splitting Energy ($\Delta_0$): The energy difference between the newly split $t_{2g}$ and $e_g$ sets is denoted by $\Delta_0$ (or 10 Dq).

2. Limitations of Crystal Field Theory (CFT):

• Ignores Covalent Character: CFT assumes purely ionic/electrostatic bonds between the metal and ligand. It completely fails to explain the partial covalent character observed in coordination complexes (e.g., evidenced by EPR spectroscopy and nephelauxetic effect).
• Fails to Explain Ligand Strength (Spectrochemical Series): Because ligands are treated merely as point charges, an anionic ligand (like $OH^-$) should exert a stronger repulsive field than a neutral molecule (like $CO$ or $NH_3$). However, the spectrochemical series shows $CO > NH_3 > OH^-$. CFT cannot explain why neutral ligands can act as strong field ligands.
• Ignores $\pi$-bonding: CFT only accounts for $\sigma$-interaction and does not consider $\pi$-bonding (back-bonding), which is crucial for ligands like CO and $CN^-$.
• Cannot Explain Charge Transfer Spectra: Intense colors resulting from Ligand-to-Metal Charge Transfer (LMCT) cannot be explained by d-d transitions alone.

Detailed Solution:
Derivation of the Gibbs-Helmholtz Equation:
The Gibbs free energy (G) is defined thermodynamically as:
Where $H$ is enthalpy, $T$ is absolute temperature, and $S$ is entropy.

Taking the complete differential of Equation 1 yields:
$dG = dH - TdS - SdT$

We know that $H = E + PV$. Taking its differential gives:
$dH = dE + PdV + VdP$

Substitute $dH$ into the $dG$ equation:
$dG = dE + PdV + VdP - TdS - SdT$           (Equation 2)

From the First and Second Laws of Thermodynamics for a reversible process, we know $TdS = dE + PdV$.
Substituting this into Equation 2 cancels out $dE$ and $PdV$:
$dG = (TdS) + VdP - TdS - SdT$
$dG = VdP - SdT$

If the process occurs at constant pressure ($dP = 0$), the equation simplifies to:
$dG = -SdT$
Or, expressing it as a partial derivative with respect to temperature at constant pressure:
Now, returning to our original definition $G = H - TS$, we rearrange it to solve for $S$:
$-S = \frac{G - H}{T}$

Substitute Equation 3 into this rearranged expression:
$\left(\frac{\partial G}{\partial T}\right)_P = \frac{G - H}{T}$
Multiply both sides by $T$:
$T \left(\frac{\partial G}{\partial T}\right)_P = G - H$

Rearranging for $H$ yields the standard form of the Gibbs-Helmholtz equation:

Application of the Gibbs-Helmholtz Equation:

• Calculating Enthalpy Change ($\Delta H$) of a Cell Reaction: In electrochemistry, $\Delta G$ is related to EMF by $\Delta G = -nFE$. Using the Gibbs-Helmholtz equation, we can determine the Enthalpy of reaction ($\Delta H$) simply by measuring the EMF of a cell ($E$) and its temperature coefficient $\left(\frac{\partial E}{\partial T}\right)_P$.

Detailed Solution:
When a solid material (typically a ductile metal like mild steel) is subjected to an increasing tensile load, its deformation is plotted on a Stress-Strain curve. Stress ($\sigma = F/A$) is plotted on the Y-axis, and Strain ($\epsilon = \Delta L / L$) is plotted on the X-axis.

Key Points on the Stress-Strain Curve:

• Proportional Limit: From the origin to this point, the curve is a perfectly straight line. The solid rigidly obeys Hooke's Law (Stress $\propto$ Strain). The slope of this line determines Young's Modulus ($E$).
• Elastic Limit (Yield Point): The maximum stress the material can endure and still return to its original length when the load is removed. The region from origin to this point is the Elastic Region. If stretched beyond this, permanent deformation occurs.
• Upper and Lower Yield Points: In materials like mild steel, there is a distinct yield point where the material begins to deform plastically without any substantial increase in stress. This marks the onset of the Plastic Region.
• Ultimate Tensile Strength (UTS): The maximum stress (highest point on the curve) the material can withstand before necking begins. Past this point, the cross-sectional area of the specimen rapidly decreases.
• Fracture/Breaking Point: The point at which the material actually physically ruptures or snaps. The stress here is known as the breaking stress.