Which of the following relations is incorrect ( `l =` length, a = area) ?

Options

A. `Lambda = kV` (V = dilution of the solution mL/equiv. or mL/mol)
B. `G = k (a)/(l) ohm`
C. `G = k (a)/(l) ohm^(-1)`
D. `R = (1)/(k)(a)/(l) ohm`

Explanation

To determine which of the given relations is incorrect, we need to analyze each statement carefully. Let's break it down step by step.

### Step 1: Analyze the first relation
The first relation states:
\[ \Lambda_m = K \times V \]
Where:
- \(\Lambda_m\) is the molar conductivity,
- \(K\) is the conductivity,
- \(V\) is the volume.

**Understanding Molar Conductivity:**
Molar conductivity (\(\Lambda_m\)) is defined as:
\[ \Lambda_m = \frac{K}{C} \]
Where \(C\) is the concentration (in mol/L).

**Rearranging the equation:**
From the definition, we can express conductivity (\(K\)) in terms of molar conductivity and concentration:
\[ K = \Lambda_m \times C \]

If we consider volume (\(V\)), we can relate concentration and volume:
\[ C = \frac{n}{V} \]
Where \(n\) is the number of moles.

Thus, substituting \(C\) into the equation for \(K\):
\[ K = \Lambda_m \times \frac{n}{V} \]

This shows that the first relation is indeed correct.

### Step 2: Analyze the second relation
The second relation states:
\[ G = \frac{K \times A}{L} \]
Where:
- \(G\) is the conductivity,
- \(K\) is the resistivity,
- \(A\) is the area,
- \(L\) is the length.

**Understanding Conductivity:**
Conductivity (\(G\)) is the reciprocal of resistivity (\(\rho\)):
\[ G = \frac{1}{\rho} \]

**Relating Resistivity to Resistance:**
Resistivity can also be expressed in terms of resistance (\(R\)):
\[ \rho = R \times \frac{A}{L} \]
Thus,
\[ G = \frac{1}{R \times \frac{A}{L}} = \frac{L}{R \times A} \]

This means that the correct relation should be:
\[ G = \frac{L}{R \times A} \]

However, the relation given in the question is:
\[ G = \frac{K \times A}{L} \]
This indicates that \(K\) is being incorrectly used in place of resistivity.

### Conclusion:
The second relation is incorrect because it misrepresents the relationship between conductivity, area, and length. Therefore, the incorrect statement is:
\[ G = \frac{K \times A}{L} \]

### Final Answer:
The incorrect relation is:
**Option B: \(G = \frac{K \times A}{L}\)**

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