Same Capillary Rise for Different Liquids: Density and Radius Link
Physics · Mechanical Properties Of Fluids · NEET
Capillary rise is h = 2Tcos(theta)/(rho g r). So the height depends on four things: surface tension T, angle of contact theta, density rho, and tube radius r. Two different liquids can rise to the SAME height if the changes cancel out. Memory hook: "h likes small rho and small r." Heavier liquid or wider tube pulls the height down; you must raise T or cos(theta) to get it back.
Three identical capillaries reach the same height h. With T and r fixed, the denser liquid must have the smaller angle of contact, so cos(theta) is directly proportional to density.
Your doubts, answered
If two liquids rise to the same height, does that mean they have the same surface tension?
No. Same height only means the whole group 2Tcos(theta)/(rho g r) is equal. Any of T, theta, rho or r can be different, as long as they cancel out. For example a liquid with high T can still match a liquid with low T if the first one also has a larger density or a smaller cos(theta). Never assume one property is equal just because the heights match.
A denser liquid, does it rise higher or lower in the same tube?
Lower, if everything else (T, theta, r) is the same. In h = 2Tcos(theta)/(rho g r), density rho is in the bottom. So bigger rho gives smaller h. This is why a heavy liquid needs a stronger pull (higher T or a more concave meniscus) just to reach the same height as a lighter liquid.
Same height in the same tube, then rho times cos(theta) is fixed. How?
Take h = 2Tcos(theta)/(rho g r). If h, T, r and g are the same for both liquids, then cos(theta)/rho must be the same. So cos(theta) is directly proportional to rho. A denser liquid must have a LARGER cos(theta), which means a SMALLER angle of contact. This is exactly what the NEET 2016 PYQ tests.
If I make the tube radius half, what happens to the rise?
The rise doubles. Radius r is in the denominator, so h is inversely proportional to r. Thin tubes (small r) give big rise; that is why 'capilla' means hair. So to keep the SAME height in a thinner tube, you would need to change another term, for example use a denser liquid or a liquid with lower surface tension.
Does the angle of contact change the height a lot?
Yes, through cos(theta). For water on clean glass theta is near 0, so cos(theta) is near 1 and the rise is maximum. As theta grows toward 90 degrees, cos(theta) drops toward 0 and the rise falls. If theta is above 90 degrees (like mercury on glass), cos(theta) is negative and the liquid goes DOWN instead of up.
⚠️ The NEET trap ✗ Two liquids reach the same height, so they must have the same density. ✓ Same height means only 2Tcos(theta)/(rho g r) is equal. Density can differ; it is balanced by differences in T, cos(theta), or tube radius r. 🧠 'Same height' is a statement about the WHOLE fraction, not about any single property. Isolate what is truly held equal (usually T, r) before you compare rho and theta.
Real NEET questions
NEET 2016 Phase 2
Three liquids of densities rho1, rho2 and rho3 (with rho1 > rho2 > rho3), having the same value of surface tension T, rise to the same height in three identical capillaries. The angles of contact theta1, theta2 and theta3 obey:
A · pi/2 > theta1 > theta2 > theta3 >= 0
B · 0 <= theta1 < theta2 < theta3 < pi/2 ✓
C · pi/2 < theta1 < theta2 < theta3 < pi
D · pi > theta1 > theta2 > theta3 > pi/2
Solution: Use h = 2Tcos(theta)/(rho g r). The tubes are identical (same r) and T is the same, and the rise h is the same for all three. So cos(theta)/rho is constant, which means cos(theta) is directly proportional to rho. Since rho1 > rho2 > rho3, we get cos(theta1) > cos(theta2) > cos(theta3). Cosine decreases as the angle grows, so theta1 < theta2 < theta3. All liquids rise (h positive), so each theta is below 90 degrees, i.e. in [0, pi/2). This matches option B.
Solved Mechanical Properties Of Fluids NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
