A) \[\sqrt{{{x}^{2}}+{{y}^{2}}}\]
B) \[\sqrt{x+y}\]
C) \[x+y\]
D) \[\frac{x+y}{2}\]
Correct Answer: A
Solution :
Given that two soap bubbles coalesce to constitute a bubble of radius z. Now from the ideal gas law, we get \[pV={{p}_{1}}{{V}_{1}}+{{p}_{2}}{{V}_{2}}\] Hence, we have \[nRT={{n}_{1}}RT+{{n}_{2}}RT\] So, \[n={{n}_{1}}+{{n}_{2}}\] Thus, we have \[{{p}_{1}}={{p}_{0}}+\frac{4T}{x},{{p}_{2}}={{p}_{0}}+\frac{4T}{y},p={{p}_{0}}+\frac{4T}{z}\] Assuming that the process is taking place in vacuum, we have \[{{p}_{1}}=\frac{4T}{x},{{p}_{2}}=\frac{4T}{y},p=\frac{4T}{z}\] Hence, \[pV={{p}_{1}}{{V}_{1}}+{{p}_{2}}{{V}_{2}}\] or \[\frac{4T}{z}\left( \frac{4}{3}\pi {{z}^{3}} \right)=\frac{4T}{x}\left( \frac{4}{3}\pi {{x}^{3}} \right)+\frac{4T}{y}\left( \frac{4}{3}\pi {{y}^{3}} \right)\] Hence \[{{z}^{2}}={{x}^{2}}+{{y}^{2}}\Rightarrow z=\sqrt{{{x}^{2}}+{{y}^{2}}}\]
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