Question:

Two soap bubbles of radii $x$ and $y$ coalesce to constitute a bubble of radius $z$. Then $z$ is equal to

Updated On: Jul 15, 2024
  • $\sqrt{x^{2}+y^{2}}$
  • $\sqrt{x+y}$
  • $x + y$
  • $\frac{x+y}{2}$
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The Correct Option is A

Solution and Explanation

Given that two soap bubbles coalesce to constitute a bubble of radius $z$. Now from the ideal gas law, we get
$p V=p_{1} V_{1}+p_{2} V_{2}$
Hence, we have
$n R T=n_{1} R T+n_{2} R T$
So, $n=n_{1}+n_{2}$
Thus, we have
$p_{1}=p_{0}+\frac{4 T}{x}$,
$p_{2}=p_{0}+\frac{4 T}{y}$,
$p=p_{0}+\frac{4 T}{z}$
Assuming that the process is taking place in vacuum, we have
Hence, $p_{1}=\frac{4 T}{x}, p_{2}=\frac{4 T}{y}, p=\frac{4 T}{z}$
$p V=p_{1} V_{1}+p_{2} V_{2}$
or $\frac{4 T}{z}\left(\frac{4}{3} \pi z^{3}\right)=\frac{4 T}{x}\left(\frac{4}{3} \pi x^{3}\right)+\frac{4 T}{y}\left(\frac{4}{3} \pi y^{3}\right)$
Hence $z^{2}=x^{2}+y^{2} $
$\Rightarrow z=\sqrt{x^{2}+y^{2}}$
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Concepts Used:

Gas Laws

The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases.

The five gas laws are:

  • Boyle’s Law, which provides a relationship between the pressure and the volume of a gas.
  • Charles’s Law, which provides a relationship between the volume occupied by a gas and the absolute temperature.
  • Gay-Lussac’s Law, which provides a relationship between the pressure exerted by a gas on the walls of its container and the absolute temperature associated with the gas.
  • Avogadro’s Law, which provides a relationship between the volume occupied by a gas and the amount of gaseous substance.
  • The Combined Gas Law (or the Ideal Gas Law), which can be obtained by combining the four laws listed above.