A prototype sports ball for a new game is precisely inflated to a pressure of 7.3 psi at 25°C. The ball is spherical with an inner bladder diameter of 22.8 cm. It is filled with air (Mw = 28.97 g/mol). Note Avogadro's number is 6.022 x 1023 atom/mol. ba (a) calculate the number of moles of air in the ball at the specified temperature and pressure. Consider the air in the ball to follow the ideal gas law. PV = nRT 1 hanol and water of the ball Tis the absolute
A prototype sports ball for a new game is precisely inflated to a pressure of 7.3 psi at 25°C. The ball is spherical with an inner bladder diameter of 22.8 cm. It is filled with air (Mw = 28.97 g/mol). Note Avogadro's number is 6.022 x 1023 atom/mol. ba (a) calculate the number of moles of air in the ball at the specified temperature and pressure. Consider the air in the ball to follow the ideal gas law. PV = nRT 1 hanol and water of the ball Tis the absolute
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
Related questions
Question
![**Problem Statement:**
(b) How many oxygen atoms compose the air in the ball? Assume that the mole fraction of \( \text{N}_2 \) in the air is 0.79 and the remainder of the air is \( \text{O}_2 \).
---
**Analysis:**
To solve this problem, we will determine the mole fraction of oxygen gas (\( \text{O}_2 \)) in the air. Given that nitrogen gas (\( \text{N}_2 \)) has a mole fraction of 0.79, the remaining portion of the air is composed of oxygen gas. Knowing that air is a mixture, the mole fraction can be calculated as follows:
**Step-by-Step Calculation:**
1. **Calculate the Mole Fraction of \( \text{O}_2 \):**
The mole fraction of \( \text{O}_2 \) can be determined by subtracting the mole fraction of \( \text{N}_2 \) from 1.
\[
\text{Mole fraction of } \text{O}_2 = 1 - 0.79 = 0.21
\]
2. **Calculate the Number of Oxygen Atoms:**
Since molecular oxygen (\( \text{O}_2 \)) is diatomic, each molecule contains 2 oxygen atoms. If \( n \) is the number of moles of \( \text{O}_2 \), then the number of oxygen atoms can be found using Avogadro's number, \( 6.022 \times 10^{23} \text{ atoms/mole} \).
\[
\text{Number of oxygen atoms} = n \times 2 \times 6.022 \times 10^{23}
\]
3. **Example:**
Assume there are a certain number of moles of air in the ball, you can then multiply the moles of \( \text{O}_2 \) (0.21 times the total moles of air) by \( 2 \times 6.022 \times 10^{23} \) to get the total number of oxygen atoms.
---
**Conclusion:**
By following these steps, you can compute the number of oxygen atoms in the air inside the ball based on the mole fraction of \( \text{O}_2 \) and \( \text{N}_2 \). This concept illustrates](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F731e0604-5c85-4793-9a95-bf37d5a4c7f6%2F8a333685-ebad-42f2-a643-722d108bd656%2F37wula_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
(b) How many oxygen atoms compose the air in the ball? Assume that the mole fraction of \( \text{N}_2 \) in the air is 0.79 and the remainder of the air is \( \text{O}_2 \).
---
**Analysis:**
To solve this problem, we will determine the mole fraction of oxygen gas (\( \text{O}_2 \)) in the air. Given that nitrogen gas (\( \text{N}_2 \)) has a mole fraction of 0.79, the remaining portion of the air is composed of oxygen gas. Knowing that air is a mixture, the mole fraction can be calculated as follows:
**Step-by-Step Calculation:**
1. **Calculate the Mole Fraction of \( \text{O}_2 \):**
The mole fraction of \( \text{O}_2 \) can be determined by subtracting the mole fraction of \( \text{N}_2 \) from 1.
\[
\text{Mole fraction of } \text{O}_2 = 1 - 0.79 = 0.21
\]
2. **Calculate the Number of Oxygen Atoms:**
Since molecular oxygen (\( \text{O}_2 \)) is diatomic, each molecule contains 2 oxygen atoms. If \( n \) is the number of moles of \( \text{O}_2 \), then the number of oxygen atoms can be found using Avogadro's number, \( 6.022 \times 10^{23} \text{ atoms/mole} \).
\[
\text{Number of oxygen atoms} = n \times 2 \times 6.022 \times 10^{23}
\]
3. **Example:**
Assume there are a certain number of moles of air in the ball, you can then multiply the moles of \( \text{O}_2 \) (0.21 times the total moles of air) by \( 2 \times 6.022 \times 10^{23} \) to get the total number of oxygen atoms.
---
**Conclusion:**
By following these steps, you can compute the number of oxygen atoms in the air inside the ball based on the mole fraction of \( \text{O}_2 \) and \( \text{N}_2 \). This concept illustrates
![**Prototype Ball Analysis Using Ideal Gas Law**
A prototype sports ball for a new game is precisely inflated to a pressure of 7.3 psi at 25°C. The ball is spherical with an inner bladder diameter of 22.8 cm. It is filled with air (Mw = 28.97 g/mol). Note Avogadro's number is \(6.022 \times 10^{23}\) atoms/mol.
**Task (a):** Calculate the number of moles of air in the ball at the specified temperature and pressure. Consider the air in the ball to follow the ideal gas law.
**Ideal Gas Law Equation:**
\[ PV = nRT \]
- \( P \) is the absolute pressure of the air in atm
- \( V \) is the volume of the ball
- \( T \) is the absolute temperature in Kelvin (K)
- \( R \) is the ideal gas constant
**Ideal Gas Constant (\( R \)) Values:**
- \( 8.314 \, \text{J mol}^{-1} \, \text{K}^{-1} \)
- \( 0.082 \, 05 \, \text{L atm mol}^{-1} \, \text{K}^{-1} \)
- \( 8.205 \, 73 \, \text{m}^3 \, \text{atm mol}^{-1} \, \text{K}^{-1} \)
- \( 1.987 \, \text{cal mol}^{-1} \, \text{K}^{-1} \)
- \( 62.363 \, \text{mmHg L mol}^{-1} \, \text{K}^{-1} \)
- \( 1 \, 545.348 \, \text{ft lb}_f \, \text{lbmol}^{-1} \, \text{K}^{-1} \)
- \( 1.985 \, \text{Btu lbmol}^{-1} \, \text{R}^{-1} \)
- \( 10.731 \, \text{ft}^3 \, \text{psi lbmol}^{-1} \, \text{R}^{-1} \)
The goal is to use the ideal gas law to find the number of moles of air (\( n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F731e0604-5c85-4793-9a95-bf37d5a4c7f6%2F8a333685-ebad-42f2-a643-722d108bd656%2Fhixwlu_processed.png&w=3840&q=75)
Transcribed Image Text:**Prototype Ball Analysis Using Ideal Gas Law**
A prototype sports ball for a new game is precisely inflated to a pressure of 7.3 psi at 25°C. The ball is spherical with an inner bladder diameter of 22.8 cm. It is filled with air (Mw = 28.97 g/mol). Note Avogadro's number is \(6.022 \times 10^{23}\) atoms/mol.
**Task (a):** Calculate the number of moles of air in the ball at the specified temperature and pressure. Consider the air in the ball to follow the ideal gas law.
**Ideal Gas Law Equation:**
\[ PV = nRT \]
- \( P \) is the absolute pressure of the air in atm
- \( V \) is the volume of the ball
- \( T \) is the absolute temperature in Kelvin (K)
- \( R \) is the ideal gas constant
**Ideal Gas Constant (\( R \)) Values:**
- \( 8.314 \, \text{J mol}^{-1} \, \text{K}^{-1} \)
- \( 0.082 \, 05 \, \text{L atm mol}^{-1} \, \text{K}^{-1} \)
- \( 8.205 \, 73 \, \text{m}^3 \, \text{atm mol}^{-1} \, \text{K}^{-1} \)
- \( 1.987 \, \text{cal mol}^{-1} \, \text{K}^{-1} \)
- \( 62.363 \, \text{mmHg L mol}^{-1} \, \text{K}^{-1} \)
- \( 1 \, 545.348 \, \text{ft lb}_f \, \text{lbmol}^{-1} \, \text{K}^{-1} \)
- \( 1.985 \, \text{Btu lbmol}^{-1} \, \text{R}^{-1} \)
- \( 10.731 \, \text{ft}^3 \, \text{psi lbmol}^{-1} \, \text{R}^{-1} \)
The goal is to use the ideal gas law to find the number of moles of air (\( n
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