Answer to Question #299742 in Physics for Autumn

Question

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.”.

Answer 1

Given:

$h=42\:\rm m$

$T_1=2.1+273=275.1\:\rm K$

$T_2=23+273=298\:\rm K$

$p_a=10^5\:\rm Pa$

The equation of state of a perfect gas says

$$\frac{p_1V_1}{T_1}=\frac{p_2V_2}{T_2}$$

Here

$$p_1=p_a+g\rho h,\quad p_2=p_a$$

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

$$\frac{V_1}{V_2}=\frac{p_2T_1}{p_1T_2}=\frac{p_aT_1}{(p_a+g\rho h)T_2}$$

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