$$ \text { Write equations for the electrolysis of } \mathrm{CaH}_{2} \text { in fused state. } $$

Understanding the process of electrolysis begins with the concept of the dissociation of electrolytes. An electrolyte is a substance that produces ions when dissolved or melted, thereby becoming capable of conducting electricity. In our case, \textbf{calcium hydride} (\text{CaH}\(_2\)), when in a fused state, dissociates into calcium cations (\text{Ca}\(^{2+}\)) and hydride anions (H\(^-\)).

This dissociation is crucial as it provides the charged particles necessary for the electrolytic reactions. In the molten state, the lack of a rigid structure allows these ions to move freely and carry electrical current through the liquid. The equation representing this is: \[ \text{CaH}_2 \rightarrow \text{Ca}^{2+} + 2\text{H}^- \] The \text{Ca}\(^{2+}\) ions will migrate towards the cathode, while the H\(^-\) ions will move towards the anode.

During electrolysis, the electrodes, made of inert substances like platinum or graphite, play an essential role. Reactions at these electrodes are divided into two halves: the cathode reaction and the anode reaction, where reduction and oxidation occur, respectively.

At the \textbf{cathode}, which is the negative electrode, calcium ions (\text{Ca}\(^{2+}\)) gain electrons (e\(^-\)) to form solid calcium (Ca). This process is reduction because it involves a decrease in oxidation number. The cathode reaction can be represented as: \[ \text{Ca}^{2+} + 2\text{e}^- \rightarrow \text{Ca} \] Conversely, at the \textbf{anode}, the positive electrode, hydride ions (H\(^-\)) lose electrons to become hydrogen gas (\text{H}\(_2\)). Each hydride ion donates one electron, and two hydride ions are necessary to produce one molecule of hydrogen gas. This part of the process is called oxidation. The anode reaction is: \[ 2\text{H}^- \rightarrow \text{H}_2 + 2\text{e}^- \]

Each electrode reaction during electrolysis can be characterized as a half-reaction. These half-reactions showcase the loss or gain of electrons, a defining trait of redox processes. The half-reactions occurring in the electrolysis of \text{CaH}\(_{2}\) highlight this electron transfer clearly.

\textbf{At the cathode:} \[ \text{Ca}^{2+} + 2\text{e}^- \rightarrow \text{Ca} \] This half-reaction shows the reduction of calcium ions, which gain electrons to form neutral calcium metal.

\textbf{At the anode:} \[ 2\text{H}^- \rightarrow \text{H}_2 + 2\text{e}^- \] Here we see the oxidation of hydride ions, which release electrons and result in the formation of hydrogen gas.

For the process to be continuous, the number of electrons lost must equal the number of electrons gained, maintaining the electrical neutrality of the system. Hence, the ability to balance these half-reactions and ensure that the electrons cancel each other out is vital.

Quantitative description of electrolysis involves writing out balanced electrolysis equations. These equations encompass the entire reaction happening within the electrolytic cell, combining the cathode and anode reactions.

The balanced formula for the electrolysis of molten \text{CaH}\(_{2}\) outlines the complete process, starting from dissociation to the final production of elements. For every mole of \text{CaH}\(_{2}\) that undergoes electrolysis, one mole of calcium and one mole of hydrogen gas is generated. The overall balanced equation is succinctly expressed as: \[ \text{CaH}_2 \rightarrow \text{Ca} + \text{H}_2 \] It's vital to recognize that the electrons involved in the half-reactions must be balanced, allowing for the correct stoichiometry of the full electrolysis equation. This demonstrates conservation of mass and charge, a fundamental principle in chemistry.

This comprehensive formulation encapsulates the entirety of the electrolytic process, embodying the start-to-finish transformation within the cell.