The user wants me to identify the correct statements regarding the thermal dissociation of calcium carbonate: $CaCO_3$(s) $\leftrightharpoons$ $CaO$(s) + $CO_2$(g).

Let's analyze each statement step-by-step.

Analysis of the Equilibrium

The given reaction is a heterogeneous equilibrium. The equilibrium constant expression involves the concentrations (or partial pressures) of the gaseous species and the activities of the pure solid/liquid species. The activity of a pure solid is taken as 1.

For the reaction $CaCO_3$(s) $\leftrightharpoons$ $CaO$(s) + $CO_2$(g):

• The equilibrium constant in terms of partial pressures is $K_p = \frac{a_{CaO(s)} \cdot P_{CO_2(g)}}{a_{CaCO_3(s)}}$.
• Since the activities of pure solids $CaO$ and $CaCO_3$ are unity ($a_{CaO(s)} = 1$ and $a_{CaCO_3(s)} = 1$), the expression simplifies to: $K_p = P_{CO_2}$
• The equilibrium constant $K_p$ is a function of temperature only.

Now, let's evaluate each option:

A: $\Delta H$ is dependent on T

1. The variation of the enthalpy change of a reaction ($\\\Delta H$) with temperature (T) is described by Kirchhoff's equation: $\frac{d(\\Delta H)}{dT} = \Delta C_p$ where $\Delta C_p$ is the difference in molar heat capacities at constant pressure between products and reactants: $\Delta C_p = \sum C_{p, products} - \sum C_{p, reactants}$
2. For this reaction, $\Delta C_p = C_{p,m}(CaO, s) + C_{p,m}(CO_2, g) - C_{p,m}(CaCO_3, s)$.
3. In general, the heat capacities of the substances involved are different and are themselves functions of temperature. Thus, $\Delta C_p$ is generally not zero.
4. Since $\Delta C_p \neq 0$, the enthalpy of reaction $\Delta H$ changes with temperature. So, the statement is correct.

B: K is independent of the initial amount of $CaCO_3$

1. The equilibrium constant (K) for a given reaction depends only on temperature. It is a thermodynamic property that defines the state of equilibrium.
2. The initial amount of reactant ($CaCO_3$) determines whether equilibrium can be achieved and the extent of the reaction, but it does not alter the value of K itself.
3. As long as some $CaCO_3$(s) and $CaO$(s) are present in the container at equilibrium, the pressure of $CO_2$(g) will be fixed at a specific value ($P_{CO_2} = K_p$) at a given temperature, regardless of the initial amounts of the solids.
4. Therefore, the statement that K is independent of the initial amount of $CaCO_3$ is correct.

C: K is dependent on the pressure of $CO_2$ at a given T

1. This statement is causally incorrect. The equilibrium constant K is a constant value at a given temperature (T).
2. The value of K determines what the equilibrium pressure of $CO_2$ must be. It is not the other way around. At a given temperature T, the system adjusts itself until the partial pressure of $CO_2$ becomes equal to the pre-determined value of $K_p$.
3. If the pressure of $CO_2$ is changed (e.g., by adding more $CO_2$), the equilibrium will shift according to Le Chatelier's principle, but the value of K remains unchanged. The system will shift to the left to consume the added $CO_2$ until its pressure returns to the equilibrium value, $K_p$.
4. Thus, K is independent of the pressure of $CO_2$. So, the statement is incorrect.

D: $\Delta H$ is independent of catalyst, if any

1. A catalyst provides an alternative reaction pathway with a lower activation energy. It increases the rate of both the forward and reverse reactions, allowing the system to reach equilibrium faster.
2. However, a catalyst does not affect the thermodynamic properties of the reaction, such as the enthalpy of reactants and products.
3. The change in enthalpy, $\Delta H = H_{products} - H_{reactants}$, is a state function. It depends only on the initial (reactants) and final (products) states, not on the path taken to get from one to the other.
4. Since the catalyst does not change the enthalpies of the reactants or products, it does not change $\Delta H$. Therefore, the statement is correct.

Conclusion: The correct statements are A, B, and D.