Volume of a bubble rising in a lake

In summary, an air bubble initially at the bottom of a lake with a volume of 20 cm^3 and a temperature of 4.0°C rises to the surface with a temperature of 20°C. Assuming the temperature of the bubble's air is the same as the surrounding water, the change in volume can be calculated using the equations PV=kT and ΔP=40ρg. The initial and final pressures are P_{atm}+40ρg and P_{atm} respectively, resulting in a final volume of V_f=V_i*(T_f/T_i)*(P_{atm}+40ρg)/P_{atm}.
  • #1
Nathanael
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Homework Statement


An air bubble of volume 20 cm^3 is at the bottom of a lake 40 m deep, where the temperature is 4.0°C. The bubble rises to the surface, which is at a temperature of 20°C. Take the temperature of the bubble’s air to be the same as that of the surrounding water. Just as the bubble reaches the surface, what is its volume?

Homework Equations


[itex]PV=kT[/itex]
[itex]ΔP=40ρg[/itex]

The Attempt at a Solution


I don't understand how the answer doesn't depend on the atmospheric pressure. If the atmospheric pressure were greater, then wouldn't the change in volume be smaller?
 
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  • #2
Nathanael said:
bottom of a lake 40 m deep,
Nathanael said:
rises to the surface
 
  • #3
Right... But that only gives you the difference in pressure. Isn't the atmospheric pressure still relevant?
 
  • #4
Yes. What are the initial and final pressures? Don't feel that adding the "stack" is an unjustified ad hoc step to take. Pressure is the result of the sum of all masses above a certain point x "g."
 
  • #5
Initial pressure would be [itex]P_{atm}+40ρg[/itex] (where ρ is the density of water) and the final pressure would be [itex]P_{atm}[/itex]

This gives me a final volume of [itex]V_f=V_i(\frac{T_f}{T_i})(\frac{P_{atm}+40ρg}{P_{atm}})[/itex]
 
  • #6
That's the way to play it. Hopefully whoever wrote the problem remembered it the same way.
 

Related to Volume of a bubble rising in a lake

What is the volume of a bubble rising in a lake?

The volume of a bubble rising in a lake can vary greatly depending on several factors such as the size and depth of the lake, the temperature and pressure of the water, and the composition of the gas inside the bubble. Therefore, it is difficult to give a specific volume for a bubble rising in a lake.

How is the volume of a bubble determined?

The volume of a bubble is determined by its size and the amount of gas inside it. The larger the bubble and the higher the gas content, the greater the volume will be. Other factors, such as the temperature and pressure of the surrounding water, can also affect the volume of a bubble.

Does the volume of a bubble change as it rises in a lake?

Yes, the volume of a bubble will change as it rises in a lake. This is because the pressure and temperature of the water will decrease as the bubble rises, causing it to expand and increase in volume.

Can the volume of a bubble rising in a lake be accurately measured?

It is difficult to accurately measure the volume of a bubble rising in a lake due to the constantly changing conditions of the water. However, scientists have developed methods to estimate the volume by taking into account the size and composition of the bubble, as well as the water temperature and pressure.

Why is the volume of a bubble rising in a lake important to study?

The volume of a bubble rising in a lake is important to study because it can provide insights into the physical and chemical processes occurring in the lake. For example, changes in the volume of bubbles can indicate changes in the water temperature and pressure, which can affect the health of aquatic ecosystems.

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