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Chapter 1: Q.1.15 (page 8)

Estimate the average temperature of the air inside a hot-air balloon (see Figure 1.1). Assume that the total mass of the unfilled balloon and payload is 500 kg. What is the mass of the air inside the balloon?

Short Answer

Expert verified

The mass of air inside the balloon isma=(500)ToTi

Step by step solution

01

Given information

mass of unfilled balloons and payload is mb=500kg

The temperature of air inside the balloon is Ti

The temperature outside the air balloon is To

molar mass of dry air is Ma

mass of air inside the balloon is ma

density of air isρa

02

Explanation

The buoyant force of the air and the gravitational pull on the balloon are in equilibrium:

FB,a=Fg,a

From which follows:

ρagV=mbg

Rewriting as an expression for volume:

V=mbρa

Using the ideal gas law in terms of density:

V=mbRToMaP

Rewriting the ideal gas law, in this case in terms of the amount of moles of air inside the balloon:

n=PVRTi=mbMaToTi

Finally, the mass of the air inside is given by the relation:

ma=mbToTi

Plugging the value of mass of balloon in the above equation, we get,

ma=(500)ToTi

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Most popular questions from this chapter

Make a rough estimate of the thermal conductivity of helium at room temperature. Discuss your result, explaining why it differs from the value for air.

Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case).

By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by

cs=Bρ,

where ρis the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness? More precisely, if we imagine applying an increase in pressure ΔPto a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:

B=ΔP–ΔV/V

This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way).

  1. Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
  2. Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
  3. Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the RMS speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
  4. When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?

In analogy with the thermal conductivity, derive an approximate formula for the viscosity of an ideal gas in terms of its density, mean free path, and average thermal speed. Show explicitly that the viscosity is independent of pressure and proportional to the square root of the temperature. Evaluate your formula numerically for air at room temperature and compare to the experimental value quoted in the text.

Calculate the rms speed of a nitrogen molecule at a room temperature?
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