If the conductivity of mercury at 0°C is 1.07 × 10^6 S m^-1 and the resistance of a cell containing

Q: Why do we multiply conductivity by resistance to get the cell constant?

Conductivity ($\kappa$) is defined as conductance ($G$) multiplied by the cell constant ($G^*$). Since conductance is the reciprocal of resistance ($\frac{1}{R}$), the equation becomes $\kappa = \frac{G^*}{R}$. Rearranging this gives $G^* = \kappa \times R$.

Q: Does the cell constant change if we use a different electrolyte instead of mercury?

No, the cell constant ($G^* = \frac{l}{A}$) depends purely on the physical geometry of the cell—specifically, the distance between the electrodes ($l$) and their cross-sectional area ($A$). It is completely independent of the liquid inside.

Q: What are the units of the cell constant?

The cell constant is calculated as length divided by area ($\frac{l}{A}$). If length is in meters and area is in square meters, the unit is $\text{m}^{-1}$. If measured in centimeters, it would be $\text{cm}^{-1}$.

Q: Why did we shift the decimal point in the final step?

The question specifically asked for the answer in the format of $x \times 10^4 \text{ m}^{-1}$. Our initial calculation gave us a value with $10^6$. To match the required format, we shifted the decimal two places to the right, converting $0.26 \times 10^6$ to $26 \times 10^4$.

Q: Can I use conductance instead of resistance for the calculation?

Absolutely! You can first calculate the conductance $G = \frac{1}{R}$, which gives $4.115 \ \Omega^{-1}$. Then, use the formula $G^* = \frac{\kappa}{G}$. Both methods are mathematically identical and will yield the exact same result.

Question

If the conductivity of mercury at $0^\circ	ext{C}$ is $1.07 	imes 10^6 	ext{ S m}^{-1}$ and the resistance of a cell containing mercury is $0.243 \ \Omega$, then the cell constant of the cell is $x 	imes 10^4 	ext{ m}^{-1}$. The value of $x$ is ......... (Nearest integer)

Sigma Percentile · Accepted Answer

Imagine you are standing in a high-tech laboratory, staring at a sleek conductivity cell filled with shimmering liquid mercury. This simple device holds the key to understanding how easily electricity can flow through different materials. In this problem, we are tasked with finding a fundamental geometric property of this cell—its **cell constant**.

### The Anatomy of a Conductivity Cell

Before we dive into the math, let's visualize what we are dealing with. A conductivity cell consists of two parallel electrodes separated by a specific distance, $l$, and each having a cross-sectional area, $A$. The ratio of this distance to the area, $\frac{l}{A}$, is known as the **cell constant**, denoted by $G^*$.

We are given two crucial pieces of information about the mercury inside the cell:
1. The **conductivity** ($\kappa$) is $1.07 	imes 10^6 	ext{ S m}^{-1}$.
2. The **resistance** ($R$) offered by the cell is $0.243 \ \Omega$.

Our mission is to find the cell constant $G^*$ and express it in the format $x 	imes 10^4 	ext{ m}^{-1}$.

### The Master Equation: Connecting the Dots

How do conductivity, resistance, and the cell constant relate to one another? The fundamental equation in electrochemistry that ties these together is:

$$\kappa = G 	imes G^*$$

Here, $G$ is the **conductance**, which is simply the reciprocal of resistance ($G = \frac{1}{R}$). Substituting this into our equation gives:

$$\kappa = \frac{1}{R} 	imes G^*$$

Rearranging this to solve for the cell constant, we get our master formula:

$$G^* = \kappa 	imes R$$

This elegant equation tells us that the cell constant is simply the product of the liquid's conductivity and the resistance it experiences in that specific cell.

### Crunching the Numbers

Now, let's bring in the values we know. We substitute $\kappa = 1.07 	imes 10^6 	ext{ S m}^{-1}$ and $R = 0.243 \ \Omega$ into our master formula:

$$G^* = (1.07 	imes 10^6) 	imes (0.243)$$

Let's perform the multiplication carefully. Multiplying $1.07$ by $0.243$ gives us $0.26001$. Therefore, our cell constant is:

$$G^* = 0.26001 	imes 10^6 	ext{ m}^{-1}$$

### The Final Polish

The question specifically asks for the answer in the format of $x 	imes 10^4 	ext{ m}^{-1}$. To achieve this, we need to shift the decimal point two places to the right, which decreases the exponent of $10$ by $2$:

$$G^* = 26.001 	imes 10^4 	ext{ m}^{-1}$$

Comparing this to the required format $x 	imes 10^4 	ext{ m}^{-1}$, we can see that $x = 26.001$. Since the question asks for the nearest integer, we round this value to get our final answer:

**$x = 26$**

And there you have it! By understanding the relationship between conductivity, resistance, and the physical geometry of the cell, we've successfully decoded the cell constant.

Enter Numerical Value:

Visualized Solution

Visualizing the Setup

Formula for Cell Constant $G^{*}$

Substituting $κ$ and $R$

Calculating $G^{*}$

Finding the value of $x$

The Way Forward

The Sigma Insight: Electrolytic Conduction

The Anatomy of a Conductivity Cell

The Master Equation

Crunching the Numbers

The Final Polish

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