   # The total mass that can be lifted by a balloon is given by the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon. Consider a hot-air balloon that approximates a sphere 5.00 m in diameter and contains air heated to 65°C. The surrounding air temperature is 21°C. The pressure in the balloon is equal to the atmospheric pressure, which is 745 torr. a. What total mass can the balloon lift? Assume that the average molar mass of air is 29.0 g/mol. ( Hint: Heated air is less dense than cool air.) b. If the balloon is filled with enough helium at 21°C and 745 torr to achieve the same volume as in part a, what total mass can the balloon lift? c. What mass could the hot-air balloon in part a lift if it were on the ground in Denver, Colorado, where a typical atmospheric pressure is 630. torr? ### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243

#### Solutions

Chapter
Section ### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243
Chapter 8, Problem 157CP
Textbook Problem
20 views

## The total mass that can be lifted by a balloon is given by the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon. Consider a hot-air balloon that approximates a sphere 5.00 m in diameter and contains air heated to 65°C. The surrounding air temperature is 21°C. The pressure in the balloon is equal to the atmospheric pressure, which is 745 torr.a. What total mass can the balloon lift? Assume that the average molar mass of air is 29.0 g/mol. (Hint: Heated air is less dense than cool air.)b. If the balloon is filled with enough helium at 21°C and 745 torr to achieve the same volume as in part a, what total mass can the balloon lift?c. What mass could the hot-air balloon in part a lift if it were on the ground in Denver, Colorado, where a typical atmospheric pressure is 630. torr?

Interpretation Introduction

Interpretation: For the given conditions, total mass that can be lifted by a balloon should be determined.

Concept introduction:

• Total mass that can be lifted by a balloon is the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon
• Total number of moles of gases in the mixture of gases can be determined by using ideal gas equation.

According to ideal gas equation,

Total numberofmoles=Total pressure×VolumeR×Temperature

R= 0.08206LatmKmol , gas constant

• Volume of sphere  =43πr3

### Explanation of Solution

Explanation

To determine: Mass of hot air when the average molar mass of air 29.0g/mol

Mass of hot air = 6.70×104g

Volume of sphere  =43πr3

Here in the case of balloon,

Volumeofhotair=43πr3=43π(2.50m)3=65.4m3

Since,1cm3=1000L65.4m3×(10dmm)×1Ldm3=6.54×104L

Total number of moles of gases in the mixture of gases can be determined by using ideal gas equation.

According to ideal gas equation,

Total numberofmoles=Total pressure×VolumeR×Temperature

Here, from the given data,

P=745torr=0.980atmSince,1atm=760torr745torr=745torr760torr=0.980atmV=6.54×104LT=65°C=338KSince,K=°C+273=(65°C+273)K=338KR=0.08206LatmKmoln=PVRT=0.980atm×6.54×104L0.08206LatmKmol×338K=2.31×103molairMassofhotair=numberofmolesofair×molarmassofhotairMolarmassofhotair=29.0g/molNumberofmolesofair=2.31×103molMassofhotair=2.31×103mol×29.0g/mol=6.70×104g

To determine: Mass of air displaced when the average molar mass of air 29.0g/mol

Mass of air displaced =7.71×104g

Mass of Helium

Total number of moles of gases in the mixture of gases can be determined by using ideal gas equation.

According to ideal gas equation,

Total numberofmoles=Total pressure×VolumeR×Temperature

Here, from the given data,

P=745torr=0.980atmSince,1atm=760torr745torr=745torr760torr=0.980atmV=6.54×104LT=21°C=294KSince,K=°C+273=(21°C+273)K=294KR=0.08206LatmKmoln=PVRT=0.980atm×6.54×104L0.08206LatmKmol×294K=2.66×103molairMassofhotair dispalced=numberofmolesofair×molarmassofhotairMolarmassofhotair=29.0g/molNumberofmolesofair=2.66×103molMassofairdispalced=2.66×103mol×29.0g/mol=7.71×104g

To determine: Total mass that can be lifted by a balloon, when the average molar mass of air 29.0g/mol

Total mass that can be lifted by a balloon, when the average molar mass of air 29.0g/mol = 1.01×104g

Total mass that can be lifted by a balloon is the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon.

Total mass that can be lifted by a balloon=Massofairdiaplaced-Mass ofhotair=7.71×104g-6.70×104g=1.01×104g

To determine: Total mass that can be lifted by a balloon, when the balloon is filled with helium.

Total mass that can be lifted by a balloon, when the balloon is filled with helium is 6.65×104g

• Mass of air displaced is same, that is 7.71×104g

Massofhelium=numberofmolesofhelium×molarmassof helium

Total number of moles of gases in the mixture of gases can be determined by using ideal gas equation.

According to ideal gas equation,

Total numberofmoles=Total pressure×VolumeR×Temperature

Here, from the given data,

P=745torr=0.980atmSince,1atm=760torr745torr=745torr760torr=0.980atmV=6.54×104LT=21°C=294KSince,K=°C+273=(21°C+273)K=294KR=0

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