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Link Rise Height to Radius, Contact Angle and Effective Gravity

A liquid rises in a narrow tube when it wets the glass and is pushed down when it does not. Every capillary question uses one formula and asks you to vary the radius, the angle of contact, the liquid or the effective gravity.

NSEJS · Properties of Matter and Fluid Mechanics · 7 linked questions · 100% coverage

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Chemistry questions first require the material, particles or reaction condition to be identified before a formula or rule is selected.

Observation · What the stem states or showsCondition · The factor that controls the change

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The rise formula

Height of rise

h = 2 T cos θ / (r ρ g), so it is inversely proportional to the tube radius and directly proportional to cos θ

For water with

T = 75 x 10⁻³ N/m rising 0.030 m (θ = 0), the radius is r = 2 T / (h ρ g) = 0.15 / (0.03 x 1000 x 10) = 0.5 mm

A liquid with an acute angle of contact (θ < 90 degrees) rises with a concave meniscus; an obtuse angle gives a fall (depression) and a convex meniscus. For a liquid with a zero angle of contact the meniscus is hemispherical.

Weight supported and potential energy

The surface tension force on the circumference supports the column: weight

supported = 2π r T cos θ

A capillary that supports 6.28 x 10^-4 N of liquid with

T = 5 x 10⁻² N/m therefore has r = W / (2π T) = 2 x 10⁻³ m

The potential energy of the raised liquid is proportional to (r h)², which is constant because h is proportional to 1 / r, so doubling the radius leaves the potential energy of the raised water unchanged: the ratio is 1 : 1.

Comparing liquids and tilted tubes

If three liquids with the same surface tension rise to the same height in identical tubes, ρ cos θ is the same for all, so the liquid with the highest density has the largest cos θ and the smallest contact angle: θ1 < θ2 < θ3 < 90 degrees for ρ1 > ρ2 > ρ3. When a tube is tilted at an angle α from the vertical, the vertical height h stays the same but the length of the liquid column along the tube is l = h / cos α, so l1 / l2 = cos 60 / cos 37 = 5/8. In a freely falling lift, the effective g is zero, so the liquid fills the whole tube and the column length equals the tube length.

OlympIQ exam lens

✓h = 2 T cos θ / (r ρ g); h is proportional to 1 / r and to cos θ
✓θ < 90 degrees: rise and concave meniscus; θ > 90 degrees: depression and convex meniscus
✓Weight supported = 2π r T cos θ
✓For equal rise with the same tension, larger density needs a smaller contact angle
✓Tilted tube: length of column = h / cos(angle from the vertical); free fall means g_eff = 0 and the column fills the tube

OlympIQ trap

Do not measure the height of rise along a tilted tube. The vertical height is fixed by the formula; the length you see along an inclined tube is that height divided by the cosine of the angle from the vertical, so a more tilted tube holds a longer column.

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