Understanding Capillary Rise and Tube Size

Capillary rise is the phenomenon where a liquid spontaneously rises up a narrow tube, overcoming gravity. This happens because of the interplay between adhesive forces (liquid to tube wall) and cohesive forces (within the liquid), which create surface tension. The effect is more pronounced in narrower tubes.

The height ($h$) to which a liquid rises in a capillary tube is determined by the tube's radius ($r$), the liquid's properties (surface tension $\gamma$, density $\rho$), and the angle of contact ($\theta$) between the liquid and the tube wall. The standard formula is:

$h = \frac{2\gamma \cos\theta}{\rho g r}$

Here's what each symbol represents:

• $h$: The height of the capillary rise (in meters).
• $\gamma$: The surface tension of the liquid (in N/m).
• $\theta$: The contact angle (in degrees), which depends on the liquid and the solid surface.
• $\rho$: The density of the liquid (in kg/m³).
• $g$: The acceleration due to gravity (in m/s²).
• $r$: The inner radius of the tube (in meters).

Calculating Minimum Glass Tubing Size

The problem requires us to find the minimum diameter (or size) of a glass tube so that the capillary rise of water does not exceed 0.25 cm. This means we need to find the radius corresponding to the maximum allowed height ($h_{max}$).

Provided and Assumed Values:

• Maximum allowed capillary rise, $h_{max} = 0.25 \text{ cm}$
• Surface tension of water, $\gamma = 0.0075 \text{ kg(f)/m}$
• Contact angle for water on glass, $\theta \approx 0^\circ$. Thus, $\cos\theta = \cos(0^\circ) = 1$.
• Density of water, $\rho \approx 1000 \text{ kg/m}^3$.
• Acceleration due to gravity, $g \approx 9.81 \text{ m/s}^2$.

Unit Conversions for Calculation:

To use the formula consistently, we need to convert all values to SI units (meters).

• Convert maximum height to meters: $h_{max} = 0.25 \text{ cm} = 0.0025 \text{ m}$.
• Convert surface tension from kg(f)/m to N/m. Since $1 \text{ kg(f)} = 9.81 \text{ N}$: $\gamma = 0.0075 \times 9.81 \text{ N/m} = 0.073575 \text{ N/m}$.

Step-by-Step Calculation:

We need to find the minimum radius ($r_{min}$) that corresponds to $h_{max}$. Rearranging the capillary rise formula to solve for $r$:

$r = \frac{2\gamma \cos\theta}{\rho g h}$

Now, plug in the values for the maximum allowed height:

$r_{min} = \frac{2 \times \gamma \times \cos\theta}{\rho \times g \times h_{max}}$

Substitute the converted values:

$r_{min} = \frac{2 \times (0.073575 \text{ N/m}) \times 1}{1000 \text{ kg/m}^3 \times 9.81 \text{ m/s}^2 \times 0.0025 \text{ m}}$

Calculate the denominator:

$1000 \times 9.81 \times 0.0025 = 24.525 \text{ kg/(m}\cdot\text{s}^2)$

Calculate the numerator:

$2 \times 0.073575 = 0.14715 \text{ N/m}$

Now, find $r_{min}$:

$r_{min} = \frac{0.14715 \text{ N/m}}{24.525 \text{ kg/(m}\cdot\text{s}^2)} \approx 0.006 \text{ m}$

(Note: Units simplify correctly to meters, as $N = kg \cdot m/s^2$)

Determining the Minimum Diameter:

The "size" of the tubing typically refers to its diameter ($d$). The diameter is twice the radius ($d = 2r$).

$d_{min} = 2 \times r_{min}$

$d_{min} = 2 \times 0.006 \text{ m}$

$d_{min} = 0.012 \text{ m}$

Final Answer Conversion:

Convert the minimum diameter back into centimeters:

$d_{min} = 0.012 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 1.2 \text{ cm}$

Thus, the minimum size of glass tubing required to ensure the capillary rise does not exceed 0.25 cm is 1.2 cm.