Answer to Question #299742 in Physics for Autumn

Question #299742

Gray and Juvia casually vibe with each other at the bottom of a 42 m deep lake. It is 2.1°C where they currently are, and 23 °C at the surface. Find the ratio of the volume of one of Juvia’s air bubbles at her current depth to its volume when it reaches the surface.”.


1
Expert's answer
2022-02-19T04:56:01-0500

Given:

h=42mh=42\:\rm m

T1=2.1+273=275.1KT_1=2.1+273=275.1\:\rm K

T2=23+273=298KT_2=23+273=298\:\rm K

pa=105Pap_a=10^5\:\rm Pa


The equation of state of a perfect gas says

p1V1T1=p2V2T2\frac{p_1V_1}{T_1}=\frac{p_2V_2}{T_2}

Here

p1=pa+gρh,p2=pap_1=p_a+g\rho h,\quad p_2=p_a

The ratio of the volume of one of Juvia’s air bubbles at her current depth to its volume when it reaches the surface

V1V2=p2T1p1T2=paT1(pa+gρh)T2\frac{V_1}{V_2}=\frac{p_2T_1}{p_1T_2}=\frac{p_aT_1}{(p_a+g\rho h)T_2}

V1V2=105275.1(105+9.8100045)298=0.17\frac{V_1}{V_2}=\frac{10^5*275.1}{(10^5+9.8*1000*45)*298}=0.17


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