The resistance of the solution 'A' is 40 ohm and that of solution 'B' is 70 ohm, both solutions being taken in the same conductivity cell. If equal volumnsof solutions 'A' and 'B' are mixed,what will be the resistance of the mixture using the same cell?(Assume that there is no increase in the degree of dissociation of 'A' and 'B' on mixing.)

Options

A. 50.9 ohm
B. 101.8 ohm
C. 110 ohm
D. 50 ohm

Answer

To solve the problem of finding the resistance of the mixture of solutions A and B, we can follow these steps:

### Step 1: Understand the Given Data
We have:
- Resistance of solution A, \( R_A = 40 \, \Omega \)
- Resistance of solution B, \( R_B = 70 \, \Omega \)

### Step 2: Define Conductivities
Let:
- \( K_1 \) be the conductivity of solution A
- \( K_2 \) be the conductivity of solution B
- The cell constant is denoted as \( X \)

### Step 3: Relate Resistance to Conductivity
The relationship between resistance and conductivity is given by:
\[ R = \frac{1}{K \cdot X} \]
Thus, we can express the conductivities as:
\[ K_1 = \frac{1}{R_A \cdot X} = \frac{1}{40 \cdot X} \]
\[ K_2 = \frac{1}{R_B \cdot X} = \frac{1}{70 \cdot X} \]

### Step 4: Mixing Equal Volumes
When equal volumes of solutions A and B are mixed, the conductivities will be diluted. The new effective conductivities will be:
\[ K_{mix} = \frac{K_1}{2} + \frac{K_2}{2} = \frac{1}{2} \left( K_1 + K_2 \right) \]

### Step 5: Substitute Conductivities
Substituting the values of \( K_1 \) and \( K_2 \):
\[ K_{mix} = \frac{1}{2} \left( \frac{1}{40 \cdot X} + \frac{1}{70 \cdot X} \right) \]

### Step 6: Find a Common Denominator
To add the fractions, we need a common denominator:
\[ K_{mix} = \frac{1}{2} \left( \frac{70 + 40}{2800 \cdot X} \right) = \frac{1}{2} \left( \frac{110}{2800 \cdot X} \right) = \frac{55}{2800 \cdot X} \]

### Step 7: Relate to Resistance of Mixture
Using the relationship \( R_{mix} = \frac{1}{K_{mix} \cdot X} \):
\[ R_{mix} = \frac{1}{\frac{55}{2800 \cdot X} \cdot X} = \frac{2800}{55} \]

### Step 8: Calculate Resistance
Now, calculate \( R_{mix} \):
\[ R_{mix} = \frac{2800}{55} \approx 50.91 \, \Omega \]

### Conclusion
Thus, the resistance of the mixture of solutions A and B is approximately \( 50.91 \, \Omega \).

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