Step 1: Identify Gas X from the Reaction
The reaction between Ammonium chloride ($\text{NH}_4\text{Cl}$) and Calcium hydroxide ($\text{Ca(OH)}_2$) is a standard laboratory method for the preparation of ammonia gas.
The balanced chemical equation is:
\[
2\text{NH}_4\text{Cl(s)} + \text{Ca(OH)}_2\text{(s)} \longrightarrow \text{CaCl}_2\text{(s)} + 2\text{H}_2\text{O(l)} + 2\text{NH}_3\text{(g)}
\]
From the reaction, the gas X produced is Ammonia ($\text{NH}_3$).
Step 2: Evaluate Statement: "X is ammonia"
As identified in Step 1, gas X is indeed ammonia.
So, this statement is Correct.
Step 3: Evaluate Statement: "Shape of X is trigonal pyramidal"
For Ammonia ($\text{NH}_3$):
The central atom is Nitrogen (N).
Nitrogen has 5 valence electrons. It forms 3 single bonds with 3 Hydrogen atoms. This leaves 5 - 3 = 2 electrons, which form 1 lone pair.
According to VSEPR (Valence Shell Electron Pair Repulsion) theory, the presence of three bond pairs and one lone pair around the central atom results in a trigonal pyramidal molecular geometry.
So, this statement is Correct.
Step 4: Evaluate Statement: "X is an electron rich hydride"
Hydrides are compounds formed between hydrogen and other elements.
Electron-rich hydrides are those in which the central atom possesses lone pair(s) of electrons in addition to forming covalent bonds with hydrogen atoms. Elements of group 15, 16, and 17 form electron-rich hydrides.
In $\text{NH}_3$, the central Nitrogen atom has one lone pair of electrons.
Therefore, ammonia is an electron-rich hydride.
So, this statement is Correct.
Step 5: Evaluate Statement: "Hybridisation of central atom in X is sp$^2$"
For Ammonia ($\text{NH}_3$):
The central Nitrogen atom has 3 bond pairs (with H) and 1 lone pair.
The steric number (number of bond pairs + number of lone pairs) = 3 + 1 = 4.
A steric number of 4 corresponds to $\text{sp}^3$ hybridization.
Thus, the hybridization of the central nitrogen atom in $\text{NH}_3$ is $\text{sp^3$}, not $\text{sp}^2$.
So, this statement is Incorrect.
Step 6: Identify the Incorrect Statement
The question asks for the incorrect statement about X. Based on our analysis, the statement "Hybridisation of central atom in X is sp$^2$" is incorrect.
Step 7: Analyze the Options
\begin{itemize}
\item Option (1): X is ammonia. (Correct statement, so not the answer)
\item Option (2): Shape of X is trigonal pyramidal. (Correct statement, so not the answer)
\item Option (3): X is an electron rich hydride. (Correct statement, so not the answer)
\item Option (4): Hybridisation of central atom in X is sp$^2$. (Incorrect statement, this is the answer)
\end{itemize}