/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Calculate the number of molecule... [FREE SOLUTION] | 91Ó°ÊÓ

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Calculate the number of molecules in a deep breath of air whose volume is \(2.25 \mathrm{~L}\) at body temperature, \(37^{\circ} \mathrm{C}\), and a pressure of 735 torr.

Short Answer

Expert verified
First, convert the temperature to Kelvin: \(T(K) = 37^{\circ} \mathrm{C} + 273.15 = 310.15 \mathrm{K}\). Convert pressure to atm: \(P = \frac{735 \mathrm{~torr}}{760 \mathrm{~torr/atm}} = 0.967 \mathrm{~atm}\). Use the Ideal Gas Law to calculate the number of moles: \(n = \frac{(0.967 \mathrm{~atm})(2.25 \mathrm{~L})}{(0.0821 \mathrm{~L~atm/K~mol})(310.15 \mathrm{~K})} = 0.089 \mathrm{~mol}\). Convert moles to molecules using Avogadro's number: \(number~of~molecules = (0.089 \mathrm{~mol})(6.022 \times 10^{23} \mathrm{molecules/mol}) = 5.36 \times 10^{22} \mathrm{molecules}\).

Step by step solution

01

Convert the given values to appropriate units

Before we can use the Ideal Gas Law equation, it's important to convert all given values into the appropriate units. The Ideal Gas Law equation uses the following unit system: Pressure (P) in atmospheres (atm), Volume (V) in liters (L), and Temperature (T) in Kelvin (K). The given values are as follows: - Volume (V) = 2.25 L - Temperature (T) = 37°C - Pressure (P) = 735 torr First, convert the temperature from Celsius to Kelvin: T(K) = T(°C) + 273.15 Second, convert pressure from torr to atm: 1 atm = 760 torr
02

Use the Ideal Gas Law equation to calculate the number of moles

The Ideal Gas Law equation is: PV = nRT Where P is the pressure (in atm), V is the volume (in L), n is the number of moles, R is the gas constant (0.0821 L atm / K mol), and T is the temperature (in K). Rearrange the equation to solve for n: n = PV / RT Plug in the values we calculated in Step 1: n = (P * V) / (R * T)
03

Convert the number of moles to the number of molecules using Avogadro's number

Now that we have the number of moles (n), we can convert that into the number of molecules using Avogadro's number, which is 6.022 x 10^23 molecules per mole. number of molecules = n * Avogadro's number Plug in the value for n calculated in Step 2: number of molecules = n * 6.022 x 10^23
04

Calculate the final answer

Calculate the number of molecules using the values obtained in the previous steps. Remember to round off your answer according to significant figures.

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Key Concepts

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

Avogadro's number
Understanding Avogadro's number is crucial for chemists and students of chemistry as it provides the bridge between macroscopic measurements and the microscopic world of atoms and molecules. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Matteo, it is a fundamental constant of physical science.

A mole is a unit of measurement in chemistry that indicates an amount of substance that contains as many elementary entities as there are atoms in exactly 12 g of carbon-12. Avogadro's number is incredibly large, being equal to approximately 6.022 x 10^23. When dealing with gases, we often use Avogadro's hypothesis, which states that at the same temperature and pressure, equal volumes of all gases contain the same number of molecules. This makes Avogadro's number a pivotal part of calculating volumes in reactions when using the Ideal Gas Law.
Gas constant
The gas constant, often denoted as R, is another cornerstone of the Ideal Gas Law. This constant provides the relationship between energy and temperature for a mole of gas particles. The value of the gas constant is 0.0821 liter-atmospheres per mole-Kelvin ( (L atm) / (mol K) ).

It's a universal value, meaning it does not change regardless of the type of gas. This constant arises from the combination of several constants: Avogadro's number and Boltzmann's constant, which refer to the energy per particle per degree Kelvin. The gas constant is pivotal in calculating the behavior of an ideal gas because it allows us to measure the internal energy involved in reactions, encapsulating both the amount of substance and the temperature in the calculations.
Molecular count
The molecular count talks about the actual number of molecules present in a substance. Leveraging from Avogadro's number, when we determine the number of moles from the Ideal Gas Law, we can readily convert that figure into the actual count of molecules by multiplying the number of moles by Avogadro's number.

This process shifts the concept from a rather abstract mole concept, which is convenient for balancing equations and understanding reactions, to a more tangible number of particles that can be used to envision the number of gas molecules involved in our everyday activities, such as taking a deep breath.
Temperature conversion
When working with gas laws, it is imperative to convert the temperature from Celsius to Kelvin. Kelvin is the base unit of temperature in the International System of Units (SI) and is used in nearly all scientific calculations. The conversion formula is simple: T(K) = T(°C) + 273.15. This equation doesn't just shift the starting point of measurement from Celsius - it reflects absolute temperatures which are essential in ensuring accurate and meaningful calculations according to the Ideal Gas Law.

Understanding how to properly convert temperature units ensures that the volume, pressure, and amount of a gas can be related in a way that accurately describes its physical behavior. This step is not to be overlooked as using the incorrect temperature scale can dramatically affect the outcome of calculations involving gases.

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Most popular questions from this chapter

A \(4.00-\mathrm{g}\) sample of a mixture of \(\mathrm{CaO}\) and \(\mathrm{BaO}\) is placed in a 1.00-L vessel containing \(\mathrm{CO}_{2}\) gas at a pressure of 730 torr and a temperature of \(25^{\circ} \mathrm{C}\). The \(\mathrm{CO}_{2}\) reacts with the \(\mathrm{CaO}\) and \(\mathrm{BaO}\), forming \(\mathrm{CaCO}_{3}\) and \(\mathrm{BaCO}_{3}\). When the reaction is complete, the pressure of the remaining \(\mathrm{CO}_{2}\) is 150 torr. (a) Calculate the number of moles of \(\mathrm{CO}_{2}\) that have reacted. (b) Calculate the mass percentage of \(\mathrm{CaO}\) in the mixture.

An open-end manometer containing mercury is connected to a container of gas, as depicted in Sample Exercise \(10.2 .\) What is the pressure of the enclosed gas in torr in each of the following situations? (a) The mercury in the arm attached to the gas is \(15.4 \mathrm{~mm}\) higher than in the one open to the atmosphere; atmospheric pressure is \(0.966\) atm. (b) The mercury in the arm attached to the gas is \(8.7 \mathrm{~mm}\) lower than in the one open to the atmosphere; atmospheric pressure is \(0.99\) atm.

Natural gas is very abundant in many Middle Eastern oil fields. However, the costs of shipping the gas to markets in other parts of the world are high because it is necessary to liquefy the gas, which is mainly methane and thus has a boiling point at atmospheric pressure of \(-164^{\circ} \mathrm{C}\). One possible strategy is to oxidize the methane to methanol, \(\mathrm{CH}_{3} \mathrm{OH}\), which has a boiling point of \(65^{\circ} \mathrm{C}\) and can therefore be shipped more readily. Suppose that \(10.7 \times 10^{9} \mathrm{ft}^{3}\) of methane at atmospheric pressure and \(25^{\circ} \mathrm{C}\) are oxidized to methanol. (a) What volume of methanol is formed if the density of \(\mathrm{CH}_{3} \mathrm{OH}\) is \(0.791 \mathrm{~g} / \mathrm{mL} ?\) (b) Write balanced chemical equations for the oxidations of methane and methanol to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\). Calculate the total enthalpy change for complete combustion of the \(10.7 \times 10^{9} \mathrm{ft}^{3}\) of methane described above and for complete combustion of the equivalent amount of methanol, as calculated in part (a). (c) Methane, when liquefied, has a density of \(0.466 \mathrm{~g} / \mathrm{mL} ;\) the density of methanol at \(25^{\circ} \mathrm{C}\) is \(0.791 \mathrm{~g} / \mathrm{mL}\). Compare the enthalpy change upon combustion of a unit volume of liquid methane and liquid methanol. From the standpoint of energy production, which substance has the higher enthalpy of combustion per unit volume?

A 6.53-g sample of a mixture of magnesium carbonate and calcium carbonate is treated with excess hydrochloric acid. The resulting reaction produces \(1.72 \mathrm{~L}\) of carbon dioxide gas at \(28^{\circ} \mathrm{C}\) and 743 torr pressure. (a) Write balanced chemical equations for the reactions that occur between hydrochloric acid and each component of the mixture. (b) Calculate the total number of moles of carbon dioxide that forms from these reactions. (c) Assuming that the reactions are complete, calculate the percentage by mass of magnesium carbonate in the mixture.

(a) Calculate the density of sulfur hexafluoride gas at 707 torr and \(21^{\circ} \mathrm{C}\). (b) Calculate the molar mass of a vapor that has a density of \(7.135 \mathrm{~g} / \mathrm{L}\) at \(12{ }^{\circ} \mathrm{C}\) and 743 torr.

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