Calculating Balloon Volume Change with Altitude

[physicsss]

A helium balloon has volume V0 and temperature T0 at sea level where the pressure is P0 and the air density is p. The balloon is allowed to float up in air to altitude y where the temperature is T1.

a) Show that the colume occupised by the balloon is then V = V0(T1/T0)e^(+cy) where c = pg/P0 = 1.25 X 10^4 m^-1.

b) Show that the buoyant force does not depend on altitude y. Assume that the skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure. [Hint] Assume that the pressure change with altitude is P = (P0)e^(+cy).

I was able to use the hint to do a), but I'm confused by b). Why does the book say to assume he skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure?

 

[Nate810]

How does this relate to the buoyant force? a) The volume occupied by the balloon is related to the temperature, pressure and density of the air. The equation for the volume change due to a change in temperature is given by V = V0(T1/T0). The equation for the volume change due to a change in pressure is given by V = V0e^(-cy), where c = pg/P0. Combining these two equations gives us the total volume change due to both pressure and temperature changes: V = V0(T1/T0)e^(-cy). b) The buoyant force is equal to the weight of the displaced air, which is equal to the mass of the displaced air times the gravitational acceleration. The mass of the displaced air is equal to the volume of the balloon times the density of the air. The density of the air is related to the pressure by the equation p = P/RT, where R is the gas constant and T is the temperature. Thus, the buoyant force can be written as Fb = Vpg/RT. Since the pressure changes with altitude (P = P0e^(+cy)), the density of the air (p = P/RT) also changes with altitude. However, if the skin of the balloon maintains the helium pressure at a constant factor of 1.05 times greater than the outside pressure, then the helium pressure inside the balloon will remain constant, thus eliminating the effect of pressure change on the buoyant force. Therefore, the buoyant force does not depend on altitude y.

 

[quicknote]

a) To show that the volume occupied by the balloon at altitude y is V = V0(T1/T0)e^(+cy), we will use the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature.

At sea level, the ideal gas law can be written as P0V0 = n0RT0, where n0 is the number of moles of gas in the balloon at sea level. We can rearrange this equation to solve for n0: n0 = P0V0/(RT0).

At altitude y, the ideal gas law can be written as PyVy = nyRT1, where ny is the number of moles of gas in the balloon at altitude y. We can rearrange this equation to solve for ny: ny = PyVy/(RT1).

Since the number of moles of gas remains constant, we can set n0 = ny and solve for Vy: Vy = (P0V0T1)/(T0Py).

Next, we can use the given value for c = pg/P0 = 1.25 X 10^4 m^-1 to substitute for Py in the equation above: Py = P0e^(cy).

Substituting this into the equation for Vy, we get Vy = (P0V0T1)/(T0P0e^(cy)) = V0(T1/T0)e^(+cy).

Therefore, the volume occupied by the balloon at altitude y is V = V0(T1/T0)e^(+cy).

b) The buoyant force on an object is equal to the weight of the fluid it displaces. In this case, the fluid is air and the object is the helium balloon. The buoyant force is given by the equation Fb = ρVg, where ρ is the density of the fluid, V is the volume of the object, and g is the acceleration due to gravity.

At sea level, the buoyant force on the balloon is Fb0 = pV0g, where p is the density of air at sea level.

At altitude y, the buoyant force on the balloon is Fby = pVyg, where p is the density of air at altitude y and Vy is the volume of the balloon at altitude y.

Substituting the equation