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Chapter 18: Problem 46

A physics lecture room at 1.00 atm and 27.0\(^\circ\)C has a volume of 216 m\(^3\). (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is N\(_2\). Calculate (b) the particle density-that is, the number of N\(_2 \) molecules per cubic centimeter-and (c) the mass of the air in the room.

Short Answer

Expert verified
Approximately 5.378 脳 10虏鈦� molecules, particle density is 2.49 脳 10鹿鈦� cm鈦宦�, and mass is 251.8 kg.

Step by step solution

01

Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin using the formula: \( T = T_{\text{Celsius}} + 273.15 \). For 27.0\(^\circ\)C, \( T = 27.0 + 273.15 = 300.15 \text{ K} \).
02

Use Ideal Gas Law to Find Moles

Apply the ideal gas law \( PV = nRT \) to find the number of moles \( n \). Rearrange the equation to solve for \( n \): \( n = \frac{PV}{RT} \). Where \( P = 1.00 \text{ atm} = 101325 \text{ Pa} \), \( V = 216 \text{ m}^3 \), \( R = 8.314 \text{ J/mol} \cdot \text{K} \), and \( T = 300.15 \text{ K} \). Substitute the values to find \( n \).
03

Calculate Number of Air Molecules

Convert moles to molecules using Avogadro's number \( 6.022 \times 10^{23} \text{ molecules/mol} \). Multiply \( n \) by Avogadro's number to obtain the total number of molecules.
04

Calculate Particle Density

Calculate the particle density by converting the volume from cubic meters to cubic centimeters \((1 \text{ m}^3 = 1 \times 10^6 \text{ cm}^3)\). Then use the formula: \( \text{Density} = \frac{\text{Number of Molecules}}{\text{Volume in cm}^3} \).
05

Calculate Mass of Air

Find the mass of the air by first calculating the molar mass of \( \text{N}_2 \), which is \( 28.02 \text{ g/mol} \). Multiply the number of moles \( n \) by the molar mass of \( \text{N}_2 \), and convert the final result from grams to kilograms (\(1 \text{ kg} = 1000 \text{ g}\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mole Calculations
When working with gases, understanding mole calculations can make complex problems easier to handle. In the context of the Ideal Gas Law, mole calculations allow us to determine how much of a substance is present in a system. The mole is a fundamental unit in chemistry that defines the amount of a substance. A mole corresponds to Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles. These could be atoms, molecules, ions or other chemical units.

For example, to estimate the number of molecules in a given volume of gas, you'll need to
  • Use the Ideal Gas Law \(PV = nRT\) to calculate the moles \(n\).
  • Here, \(P\) is pressure, \(V\) is volume, \(R\) is the ideal gas constant (8.314 J/mol路K), and \(T\) is temperature in Kelvin.
  • Once you have the number of moles, multiply by Avogadro鈥檚 number to determine the total number of molecules present.
This calculation is central to understanding how gases behave under various conditions.
Particle Density
Particle density is an important concept in understanding how densely packed molecules are within a given volume. For gases, calculating particle density can tell us how many molecules exist in a set space, usually per cubic centimeter. This could relate to the concentration of molecules and could be critical in applications like airflow and diffusion studies.

The steps to calculate particle density include:
  • Determine the total number of molecules from your mole calculations using Avogadro's number.
  • Convert the space you're measuring from cubic meters to cubic centimeters鈥攔ecall, \(1 \text{ m}^3 = 1 \times 10^6 \text{ cm}^3\).
  • Finally, use the formula \(\text{Density} = \frac{\text{Number of Molecules}}{\text{Volume in cm}^3}\) to find particle density.
This calculation can help understand how the gas might spread or react in a given environment.
Molar Mass
Molar mass plays a crucial role when transitioning from moles to real-world applications like mass. Molar mass is typically expressed in grams per mole (g/mol), and it allows us to convert between the amount of substance in moles and its mass in grams or kilograms.

In our scenario, nitrogen (\(\text{N}_2\)) is the primary component of air, with a molar mass of \(28.02 \text{ g/mol}\). To find the mass of a sample:
  • Calculate the moles \(n\) of \(\text{N}_2\) using the Ideal Gas Law.
  • Multiply the number of moles \(n\) by the molar mass of \(\text{N}_2\) to get the mass in grams.
  • If needed, convert the mass from grams to kilograms by dividing by 1000.
Understanding molar mass is essential for determining how much of a substance is present by weight, which is vital in many engineering and scientific calculations.

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Most popular questions from this chapter

Perfectly rigid containers each hold \(n\) moles of ideal gas, one being hydrogen (H\(_2\)) and the other being neon (Ne). If it takes 300 J of heat to increase the temperature of the hydrogen by 2.50\(^\circ\)C, by how many degrees will the same amount of heat raise the temperature of the neon?

A 3.00-L tank contains air at 3.00 atm and 20.0\(^\circ\)C. The tank is sealed and cooled until the pressure is 1.00 atm. (a) What is the temperature then in degrees Celsius? Assume that the volume of the tank is constant. (b) If the temperature is kept at the value found in part (a) and the gas is compressed, what is the volume when the pressure again becomes 3.00 atm?

The atmosphere of Mars is mostly CO\(_2\) (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0\(^\circ\)C in summer to -100\(^\circ\)C in winter. Over the course of a Martian year, what are the ranges of (a) the rms speeds of the CO\(_2\) molecules and (b) the density (in mol/m\(^3\)) of the atmosphere?

How many moles are in a 1.00-kg bottle of water? How many molecules? The molar mass of water is 18.0 g/mol.

The gas inside a balloon will always have a pressure nearly equal to atmospheric pressure, since that is the pressure applied to the outside of the balloon. You fill a balloon with helium (a nearly ideal gas) to a volume of 0.600 L at 19.0\(^\circ\)C. What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen (77.3 K)?
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