Question

A McLeod gauge having volume of bulbs and mercury capillary, $V = 100 \times 10^{-6}\text{ m}^3$ and the diameter of measuring capillary is $1\text{ mm}$. For the reading of $60\text{ mm}$, the pressure will be}

• A. $14\ \mu\text{m}$
• B. $28\ \mu\text{m}$
• C. $42\ \mu\text{m}$
• D. $56\ \mu\text{m}$

Hint:

Always double check units in McLeod gauge formulas.
Keep all quantities in standard SI units (meters, cubic meters) before converting to standard vacuum units like $\mu\text{m}$ of Hg (which is equivalent to milli-Torr).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The McLeod gauge is a device used to measure very low pressures (vacuum) by compressing a known volume of gas into a small capillary tube.
We need to calculate the pressure measured by the gauge for a given reading.

Step 2: Key Formula or Approach:
The formula for pressure $P$ measured by a McLeod gauge when the mercury level in the capillary is at a height $h$ (quadratic or linear scale method) is:
\[ P = \frac{a \cdot h^2}{V} \]
where:
- $a$ is the cross-sectional area of the capillary tube: $a = \frac{\pi d^2}{4}$
- $h$ is the height of the mercury column (reading)
- $V$ is the volume of the bulb and capillary

Step 3: Detailed Explanation:

• Given parameters:
Volume, $V = 100 \times 10^{-6}\text{ m}^3$.
Capillary diameter, $d = 1\text{ mm} = 10^{-3}\text{ m}$.
Mercury column height, $h = 60\text{ mm} = 0.06\text{ m}$.

• Calculate the cross-sectional area $a$:
\[ a = \frac{\pi (10^{-3})^2}{4} \approx 0.7854 \times 10^{-6}\text{ m}^2 \]

• Calculate the pressure $P$ in meters of mercury (m of Hg):
\[ P = \frac{0.7854 \times 10^{-6} \times (0.06)^2}{100 \times 10^{-6}} \]
\[ P = \frac{0.7854 \times 0.0036}{100} = 2.827 \times 10^{-5}\text{ m of Hg} \]

• Convert to micrometers of mercury ($\mu\text{m}$ of Hg):
\[ P = 2.827 \times 10^{-5}\text{ m} \times 10^6\ \mu\text{m/m} \approx 28\ \mu\text{m} \]

Step 4: Final Answer:
The pressure measured is approximately $28\ \mu\text{m}$ of Hg, which corresponds to Option (B).