Question

Let $W_1$ be the work done in blowing a soap bubble of radius $r$ from a soap solution at room temperature. The soap solution is now heated and a second soap bubble of radius $2r$ is blown from the heated soap solution. If $W_2$ is the work done in forming this second bubble, then

• A. $W_2 = 2W_1$
• B. $W_2 = 4W_1$
• C. $W_2 > 4W_1$
• D. $W_2 < 4W_1$

Hint:

If the temperature had remained constant, doubling the radius would increase the required work by a factor of $2^2 = 4$ due to the area scaling rule ($W \propto r^2$). However, because heating the solution decreases its surface tension coefficient, the required work must end up being strictly less than this unheated value, pointing directly to $W_2 < 4W_1$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Compare the work needed to form a soap bubble at two different temperatures, accounting for both radius change and surface tension variation.

Step 2: Key Formula or Approach:
Work done W ∝ T × r², where T is surface tension and r is radius. Doubling radius alone would quadruple the work (2² = 4). Heating reduces surface tension, so the actual work must be less than this purely geometric factor.

Step 3: Detailed Explanation:
If temperature were constant, doubling the radius would demand W₂ = 4W₁ due to the r² dependence of surface area. However, the heated solution has a lower surface tension coefficient. Consequently, the actual work W₂ is strictly less than 4W₁. This bounding argument—comparing against the constant-temperature baseline—narrows the answer without requiring the exact temperature coefficient of surface tension.

Step 4: Final Answer:
The work satisfies W₂<4W₁.