Concept: Molar Conductivity — For strong electrolytes, Kohlrausch's law states $\Lambda_m = \Lambda^0_m - A\sqrt{c}$, so a plot of $\Lambda_m$ vs $\sqrt{c}$ should be linear.

Step 1: Compute $\sqrt{c}$ for each concentration

Step 2: Check linearity — $\Lambda_m$ decreases uniformly as $\sqrt{c}$ increases, confirming a straight-line relationship (as $\sqrt{c}$ rises, $\Lambda_m$ falls; equivalently $\Lambda_m$ increases on dilution).

Step 3: Determine $A$ (slope) and $\Lambda^0_m$ (intercept)

Slope $= -A = \frac{147.09 - 148.61}{0.03145 - 0.01407} = \frac{-1.52}{0.01738} \approx -87.46$

So $A = 87.46\ \text{S cm}^2 \text{mol}^{-1} / (\text{mol L}^{-1})^{1/2}$.

Extending the straight line to $\sqrt{c} = 0$, the graphical intercept read from the plot is $\Lambda^0_m = 150.0\ \text{S cm}^2 \text{mol}^{-1}$.

The key idea here is Kohlrausch's law of independent migration of ions. For strong electrolytes, as you dilute the solution, ions move more freely because interionic attractions weaken. The molar conductivity $\Lambda_m$ therefore increases on dilution — and, plotted against the square root of concentration, it decreases linearly as $\sqrt{c}$ rises. This linear relationship is the hallmark of a strong electrolyte.

Why $\sqrt{c}$? Because the Debye–Hückel theory shows that the ionic atmosphere dragging on a moving ion has a radius proportional to $1/\sqrt{c}$. So the retarding effect scales with $\sqrt{c}$, and conductivity rises as $\sqrt{c}$ falls.

The equation is:

where $\Lambda_m^0$ is the limiting molar conductivity (at infinite dilution) and $A$ is a constant for the electrolyte.

Let's test this with the given data.

1. Convert the data to $\sqrt{c}$ values.

   $c$ (mol L⁻¹)	$\sqrt{c}$ (mol L⁻¹)$^{1/2}$	$\Lambda_m$ (S cm² mol⁻¹)
   0.000198	0.01407	148.61
   0.000309	0.01758	148.29
   0.000521	0.02283	147.81
   0.000989	0.03145	147.09

2. Plot $\Lambda_m$ against $\sqrt{c}$.

   The points fall on a straight line: as $\sqrt{c}$ increases, $\Lambda_m$ decreases linearly. This confirms Kohlrausch's law for KCl.

3. Find $A$ (slope magnitude).

   The slope of the line is $-A$:

So $A = 87.46\ \text{S cm}^2\text{ mol}^{-1}\text{ (mol L}^{-1})^{-1/2}$.

4. Find $\Lambda_m^0$ (the intercept). Extending the straight line to $\sqrt{c} = 0$ (infinite dilution), NCERT reads the intercept from the graph as:

Method: Kohlrausch's Law (Empirical Debye–Hückel–Onsager Plot)

Kohlrausch observed that for strong electrolytes, molar conductivity varies linearly with the square root of concentration at low concentrations:

Here:

• $\Lambda_m^0$ = limiting molar conductivity (intercept)
• $A$ = Kohlrausch constant (magnitude of the slope; the slope of the line is $-A$)

Steps

Step 1: Compute $\sqrt{c}$ for each concentration

Step 2: Plot $\Lambda_m$ vs $\sqrt{c}$

Put $\sqrt{c}$ on the x-axis and $\Lambda_m$ on the y-axis. The points fall on a straight line with negative slope.

Step 3: Determine $A$ (slope)

Take two well-separated points:

• Point 1: $(0.01407,\ 148.61)$
• Point 2: $(0.03145,\ 147.09)$

Since $\Lambda_m = \Lambda_m^0 - A\sqrt{c}$, the constant is:

Step 4: Determine $\Lambda_m^0$ (intercept)

Extend the line to $\sqrt{c} = 0$. NCERT reads the intercept from the graph as:

Final Result

• Method: Kohlrausch's empirical law (linear $\Lambda_m$ vs $\sqrt{c}$ plot)
• $\Lambda_m^0$ = 150.0 S cm² mol⁻¹
• $A$ = 87.46 S cm² mol⁻¹ (mol L⁻¹)^(−1/2)

The straight-line nature confirms KCl behaves as a strong electrolyte at these dilutions.

1. ✗ Mistake: Forgetting to convert concentration units

Students often take $c$ directly in $mol\ L^{-1}$ and then compute $c^{1/2}$ without realising that the Kohlrausch law uses $c$ in $mol\ L^{-1}$ — but the square root is fine as given.

The real trap: they forget that $\Lambda_m$ is already in $S\ cm^2\ mol^{-1}$ and try to convert it unnecessarily.

✓ How to avoid:

• Check units at the start. Here, both $c$ and $\Lambda_m$ are given in standard units.
• Only convert if the problem explicitly asks for SI units (e.g., $S\ m^2\ mol^{-1}$). For this problem, use as given.

2. ✗ Mistake: Plotting $\Lambda_m$ vs $c$ instead of $\Lambda_m$ vs $c^{1/2}$

This is the most common error. The Kohlrausch law is:

So the x-axis must be $\sqrt{c}$, not $c$.

✓ How to avoid:

• Always write the law first before plotting.
• Compute a new column: $\sqrt{c}$ for each concentration.
• Plot $\Lambda_m$ on y-axis, $\sqrt{c}$ on x-axis.

3. ✗ Mistake: Errors in calculating $\sqrt{c}$

Students sometimes:

• Take square root of the number without the unit.
• Miscalculate powers of 10 (e.g., $\sqrt{0.000198} = 0.01407$, not $0.1407$).

✓ How to avoid:

• Use scientific notation: $0.000198 = 1.98 \times 10^{-4}$ Then $\sqrt{c} = \sqrt{1.98} \times 10^{-2} \approx 1.407 \times 10^{-2}$
• Double-check each value with a calculator.

4. ✗ Mistake: Drawing a rough freehand graph and guessing intercept/slope

Students often sketch a line by eye and read $\Lambda_m^0$ from the y-intercept inaccurately.

✓ How to avoid:

• Use graph paper or plotting software.
• Draw the best-fit straight line (not just connecting dots).
• Read $\Lambda_m^0$ as the y-intercept (where $\sqrt{c} = 0$).
• Read slope $= -A$ from two far-apart points on the line.

5. ✗ Mistake: Confusing $A$ with the slope directly

The Kohlrausch law is:

So the slope of the line = $-A$. Students often take slope = $A$ and get sign wrong.

✓ How to avoid:

• Write the equation in y = mx + c form:
  • $y = \Lambda_m$
  • $x = \sqrt{c}$
  • $m = -A$
  • $c = \Lambda_m^0$
• So if slope $= -50$, then $A = 50$.

6. ✗ Mistake: Forgetting units for $\Lambda_m^0$ and $A$

Students report $\Lambda_m^0 = 150$ without units, or give $A$ in wrong units.

✓ How to avoid:

• $\Lambda_m^0$ has same units as $\Lambda_m$: $S\ cm^2\ mol^{-1}$
• $A$ has units: $S\ cm^2\ mol^{-1} \cdot (mol\ L^{-1})^{-1/2}$ (Often written as $S\ cm^2\ mol^{-1} \cdot L^{1/2}\ mol^{-1/2}$)

7. ✗ Mistake: Not checking linearity properly

Students assume the plot is a straight line without verifying.

✓ How to avoid:

• After plotting, check if points lie close to a straight line.
• For strong electrolytes like KCl, it should be linear at low concentrations.
• If one point deviates, recheck calculation of $\sqrt{c}$ for that point.

Quick Summary Table

Final tip: Before you start, write the Kohlrausch law clearly. Then compute $\sqrt{c}$, plot, find intercept ($\Lambda_m^0$) and slope ($-A$). This structured approach eliminates most errors.