You have two balloons, one large (30 cm in diameter) and one small (15 cm
in diameter) filled with helium. Both balloons are made of the same
material (latex) and have the same unfilled material thickness (2 mm) and
have a 3 cm diameter. You release both balloons at sea level and they begin
to float. As the balloon floats, the pressure that it is exposed to will
change and so will the temperature. These changes in pressure will affect
the size of the balloon. The change in the balloon size will affect the
thickness of the material, which will influence the surface tension that
the material can withstand. Plot the diameter of the balloon as a function
of height. At what height will the balloons explode? What was the velocity
of the balloon when that happened?
State your assumptions and present all of your calculations.
To answer this question, we need to make a few assumptions and consider some principles related to the behavior of balloons filled with gas and their response to changes in pressure and temperature.
Assumptions:
1. The balloons are perfect spheres.
2. The balloons are fully inflated and the gas inside them is entirely helium.
3. The effect of gravity on the balloons is negligible.
4. The balloons have enough structural integrity to withstand the surface tension forces.
Principles:
1. Boyle's Law: When the volume of a gas is reduced at constant temperature, the pressure increases.
2. Gay-Lussac's Law (also known as the pressure law): When the pressure of a gas increases, so does its temperature if the volume remains constant.
3. The Ideal Gas Law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
To plot the diameter of the balloon as a function of height, we need to consider the relationship between pressure and height. As the balloon rises, the pressure decreases due to the decrease in atmospheric pressure.
The relationship between pressure and height can be approximated by the barometric formula:
P = P₀ * exp(-M * g * h / (R * T₀)),
where P is the pressure at height h, P₀ is the initial pressure (sea level pressure), M is the molar mass of air, g is the acceleration due to gravity, R is the ideal gas constant, and T₀ is the initial temperature (sea level temperature).
Let's assume the initial sea level pressure is P₀ = 101325 Pa, the molar mass of air is M = 0.02896 kg/mol, the acceleration due to gravity is g = 9.8 m/s², and the initial sea level temperature is T₀ = 293.15 K.
We can use this formula to calculate the pressure at different heights. However, we need to be cautious because latex, the material of the balloons, is not a perfectly elastic material. As the balloon expands or contracts, the strain on the latex can cause it to thin out. We need to determine the point at which the surface tension of the material is exceeded.
To calculate the maximum pressure the balloons can withstand, we can assume that the surface tension determines the burst pressure. The burst pressure can be calculated using the equation:
P_burst = (2 * T) / r,
where P_burst is the burst pressure, T is the surface tension, and r is the radius of the balloon.
Assuming the surface tension of the balloon material is T = 0.035 N/m, we can use the radius of the balloon at any given height to calculate the burst pressure.
Now, let's calculate the diameter of the balloon as a function of height and determine when the balloons will explode.
To be able to calculate the velocity when the balloons explode, we would need to know more details regarding the shape of the bursting process and the release of the gas. Without this information, it is difficult to provide an accurate velocity estimation.