For a gaseous reaction: `A(g) rarr B(g)`, the rate expresison may be given as

Options

A. `-(d[A])/(dt) = k[A]^(n)`
B. `-(1)/(V)(dn_(A))/(dt) = k[A]^(n)`
C. `-(1)/(RT)(dP_(A))/(d t) =k[A]^(n)`
D. `-(dP)/(dt) = k[P_(A)]^(n)`

Explanation

To derive the rate expression for the gaseous reaction \( A(g) \rightarrow B(g) \), we can follow these steps:

### Step 1: Understand the Rate of Reaction
The rate of a reaction can be expressed in terms of the change in concentration of the reactant over time. For the reaction \( A \rightarrow B \), the rate can be expressed as:
\[ -\frac{d[A]}{dt} = k[A]^n \]
where:
- \( [A] \) is the concentration of reactant A,
- \( k \) is the rate constant,
- \( n \) is the order of the reaction.

### Step 2: Relate Concentration to Pressure
For gaseous reactions, we can also express the rate in terms of pressure. According to the ideal gas law, the concentration of a gas can be related to its pressure. The rate expression can be rewritten as:
\[ -\frac{dP_A}{dt} = k P_A^n \]
where \( P_A \) is the partial pressure of gas A.

### Step 3: Use Moles and Volume
We can express concentration in terms of moles and volume:
\[ [A] = \frac{n_A}{V} \]
where \( n_A \) is the number of moles of A and \( V \) is the volume. Thus, the rate can also be expressed as:
\[ -\frac{d[A]}{dt} = k \left(\frac{n_A}{V}\right)^n \]
This can be rearranged to show the relationship in terms of moles.

### Step 4: Ideal Gas Law
Using the ideal gas law \( PV = nRT \), we can express the concentration in terms of pressure:
\[ \frac{n}{V} = \frac{P}{RT} \]
Substituting this into our rate expression gives:
\[ -\frac{dP_A}{dt} = k \left(\frac{P_A}{RT}\right)^n \]
This shows how pressure relates to the rate of reaction.

### Conclusion
From the above derivations, we find that all expressions can be valid depending on how we relate concentration and pressure. Therefore, the rate expressions can be summarized as:
- \( -\frac{d[A]}{dt} = k[A]^n \)
- \( -\frac{dP_A}{dt} = k P_A^n \)
- \( -\frac{d[A]}{dt} = k \left(\frac{n_A}{V}\right)^n \)
- \( -\frac{dP_A}{dt} = k \left(\frac{P_A}{RT}\right)^n \)

Thus, all options A, B, C, and D are correct.