Question

A diver 40 m deep in 10 degrees C fresh water exhales a 1.5 cm diameter bubble.

What is the bubble's diameter just as it reaches the surface of the lake, where the water temperature is 20∘C?
Assume that the air bubble is always in thermal equilibrium with the surrounding water.

Answer 1

0.0257259766982 m

Step-by-step explanation:

`P_2` = Atmospheric pressure = 101325 Pa

`d_1` = Initial diameter = 1.5 cm

`d_2` = Final diameter

`\rho` = Density of water = 1000 kg/m³

h = Depth = 40 m

The pressure is

`P_1=P_2+\rho gh\\\Rightarrow P_1=101325+1000* 9.81* 40\\\Rightarrow P_1=493725\ Pa`

From ideal gas law we have

`(P_1V_1)/(T_1)=(P_2V_2)/(T_2)\\\Rightarrow (P_1(4)/(3*8)\pi d_1^3)/(T_1)=(P_2(4)/(3*8)\pi d_2^3)/(T_2)\\\Rightarrow (P_1d_1^3)/(T_1)=(P_2d_2^3)/(T_2)\\\Rightarrow d_2=((P_1d_1^3T_2)/(P_2T_1))^{(1)/(3)}\\\Rightarrow d_2=((493725* 0.015^3* (20+273.15))/(101325* (10+273.15)))^{(1)/(3)}\\\Rightarrow d_2=0.0257259766982\ m`

The diameter of the bubble is 0.0257259766982 m