   # A hot-air balloon is filled with air to a volume of 4.00 × 10 3 m 3 at 745 torr and 21°C. The air in the balloon is then heated to 62°C, causing the balloon to expand to a volume of 4.20 × 10 3 m 3 . What is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon? ( Hint: Openings in the balloon allow air to flow in and out. Thus the pressure in the balloon is always the same as that of the atmosphere.) ### 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 64E
Textbook Problem
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## A hot-air balloon is filled with air to a volume of 4.00 × 103 m3 at 745 torr and 21°C. The air in the balloon is then heated to 62°C, causing the balloon to expand to a volume of 4.20 × 103 m3. What is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon? (Hint: Openings in the balloon allow air to flow in and out. Thus the pressure in the balloon is always the same as that of the atmosphere.)

Interpretation Introduction

Interpretation: For the given data, the ratio of number of moles of air in the balloon should be determined.

Concept introduction:

By combining the three gaseous laws namely Boyle’s law, Charles’s law and Avogadro’s law a combined gaseous equation is obtained. This combined gaseous equation is called Ideal gas law.

According to ideal gas law,

PV=nRT

Where,

P = pressure in atmospheres

V= volumes in liters

n = number of moles

R =universal gas constant ( 0.08206L×atm/K×mol )

T = temperature in kelvins

By knowing any three of these properties, the state of a gas can be simply identified with applying the ideal gas equation. For a gas at two conditions, the unknown variable can be determined by knowing the variables that change and remain constant and can be generated an equation for unknown variable from ideal gas equation.

### Explanation of Solution

Explanation

The ratio of number of moles of air in heated balloon and normal balloon is the ratio of product of initial temperature and final volume to the product of final temperature and initial volume

According to ideal gas equation,

PV=nRT

By rearranging the above equation,

nTV=PR

Since R is a gas constant and P in this case is constant, for a gas at two conditions the equation can be written as:

n1T1V1=R=n2T2V2orn1T1V1=n2T2V2 (1)

For the given data, the ratio of number of moles

n2n1=T1V2T2V1 (2)

For finding the ratio of number of moles of air, it is needed to take and write the given data and substitute their values in the equation (2). For two conditions problem, units for P and V just needed to be the same units and it is not needed to convert the standard units. But in the case of temperature, it must be converted to the Kelvin.

T1=21°C=294Ksince,1K=°C+273=21°C+273 </

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