Question

A helium balloon is used to lift a load of 106 N. The weight of the envelope of the balloon is 51.5 N and the volume of the helium when the balloon is fully inflated is 33.9 m3. The temperature of the air is 0°C and the atmospheric pressure is 1.00 atm. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is 30.6 N. Neglect any effects due to the changes of temperature as the altitude changes. (a) How many moles of helium gas are contained in the balloon? mol (b) At what altitude will the balloon be fully inflated? km (c) Does the balloon ever reach the altitude at which it is fully inflated? Yes No (d) If the answer to Part (c) is "Yes," what is the maximum altitude attained by the balloon? (If the answer to Part (c) is "No", enter no.) km eBook

Answer 1

The buoyant force B can be found from the net upward force Fy = 30.6 N and the weights of the load Wl = 106 N and enveloper We = 51.5 N (Wl and We are the weights of the load and envelope).

This can be easily seen by a free body diagram.

B= Fy+ Wl + We

Calculate it.

The buoyant force B (calculated earlier), that the force on the helium gas by the air is given as

B=$\rho_a$ V g

where $\rho_a$ is the density of the air = 1.3 kg/m³ (at 0^o C and 1 atm) , V is the volume of the helium gas and g is the accleration due to gravity. Hence, initial V can be found using this formula.

(a) The moles of amount of helium gas can be found from the volume as such

Use the formula of ideal gas law, PV=nRT to find the number of moles (n) of helium gas.

P=1 atm (= 101325.00 N/m²), R= 8.31 J/ mol. K and T =273 K (or 0⁰ C)

(b) Now as the balloon rises the atmospheric pressure and the temperature will change, The temperature is assumed to be neglected in this case. Only the atmospheric pressure changes. The equation for change of atmospheric pressure as the altitude changes is given as, using the barometic formula

P(h)=760exp(−0.00012h)[mmHg] (where altitude at which the balloon is)

Convert the mmHg into N/m². 1 mmhg is 133.322 N/m².

Find h now as given in the problem that the volume has filly inflated (Vmax = 33.9 m³) at the height h. Use the formula

P(h)=nRT/Vmax)

(d)

This can be found by looking at the quantity

Bmax-Wl-We=Fy

where Bmax is the bouyant force at the height h using the formula $\rho_a$ Vmax g, and Wl and We are as defined earlier.

If this quantity is negative, then the balloon never reaches the height and if it is positive, then it will and the balloon will be fully inflated

(e)

use the results obtained in the above subquestions.

The maximum height can be found by considering Fy =0 and finding