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Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblem 88b
Chapter 17, Problem 88b

A helium balloon has volume V₀ and temperature T₀ at sea level where the pressure is P₀ and the air density is ρ₀. The balloon is allowed to float up in the air to altitude y where the temperature is T₁. 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 , Eq. 13–6c in Chapter 13.]

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Start by recalling the formula for the buoyant force, which is given by Archimedes' principle: F_b = ρ_air * V_displaced * g, where ρ_air is the density of the surrounding air, V_displaced is the volume of the displaced air (equal to the volume of the balloon), and g is the acceleration due to gravity.
The air density ρ_air at altitude y can be expressed using the ideal gas law: ρ_air = P / (R * T), where P is the pressure, T is the temperature, and R is the specific gas constant for air. Substitute the pressure at altitude y, P = P₀ * e⁻ᶜʸ, into this expression to get ρ_air = (P₀ * e⁻ᶜʸ) / (R * T₁).
The volume of the balloon, V, can be determined using the ideal gas law for helium. Since the pressure inside the balloon is maintained at 1.05 times the outside pressure, P_helium = 1.05 * P. Using the ideal gas law, V = (n * R_helium * T₁) / P_helium, where n is the number of moles of helium and R_helium is the specific gas constant for helium.
Substitute the expression for P_helium into the volume equation: V = (n * R_helium * T₁) / (1.05 * P₀ * e⁻ᶜʸ). Notice that the e⁻ᶜʸ term cancels out when calculating the buoyant force because it appears in both ρ_air and V.
Finally, substitute ρ_air and V into the buoyant force equation: F_b = [(P₀ / (R * T₁)) * e⁻ᶜʸ] * [(n * R_helium * T₁) / (1.05 * P₀ * e⁻ᶜʸ)] * g. Simplify the expression to show that F_b is independent of altitude y, as all altitude-dependent terms cancel out.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object submerged in it, which is equal to the weight of the fluid displaced by the object. According to Archimedes' principle, this force depends on the density of the fluid and the volume of the object submerged, not on the object's weight or the altitude at which it is located.
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Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. In the context of the helium balloon, this law helps to understand how the volume and pressure of the gas inside the balloon change with altitude and temperature, affecting the balloon's behavior in the atmosphere.
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Pressure Variation with Altitude

Pressure in the atmosphere decreases with increasing altitude due to the weight of the air above. The equation P = P₀ e⁻ᶜʸ describes this exponential decay, where P₀ is the pressure at sea level, c is a constant related to the temperature and density of the air, and y is the altitude. This relationship is crucial for understanding how the external pressure acting on the balloon changes as it rises.
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Related Practice
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