/*! 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 113 Nitrous oxide \(\left(\mathrm{N}... [FREE SOLUTION] | 91Ó°ÊÓ

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Nitrous oxide \(\left(\mathrm{N}_{2} \mathrm{O}\right)\) is a gas that is commonly used to help sedate patients in medicine and dentistry due to its mild anesthetic and analgesic properties, as well as the fact that it is nonflammable. If a cylinder of \(\mathrm{N}_{2} \mathrm{O}\) is at \(32.4 \mathrm{~atm}\) and has a volume of 5.0 \(\mathrm{L}\) at \(298 \mathrm{~K}\), how many moles of \(\mathrm{N}_{2} \mathrm{O}\) gas are in the cylinder? What volume would the gas take up if the entire contents of the cylinder were allowed to escape into a larger container that keeps the pressure constant at \(1.00 \mathrm{~atm}\) ? Assume the temperature remains at \(298 \mathrm{~K}\).

Short Answer

Expert verified
There are approximately 2.10 moles of Nâ‚‚O gas in the cylinder, and if the pressure is reduced to 1.00 atm, the gas would occupy a volume of approximately 513 L.

Step by step solution

01

Use the Ideal Gas Law formula

We will use the Ideal Gas Law formula: \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. Given: \(P = 32.4\,\text{atm}\) \(V = 5.0\,\text{L}\) \(T = 298\,\text{K}\) \(R = 0.0821\,\frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}\) Step 2: Solve the ideal gas law for the number of moles
02

Solve for n

Manipulate the Ideal Gas Law formula to solve for \(n\): \(n = \frac{PV}{RT}\) Step 3: Plug in the given values and calculate the number of moles
03

Calculate the number of moles

Substitute the given values into the formula we derived in Step 2: \(n = \frac{(32.4\,\text{atm})(5.0\,\text{L})}{(0.0821\frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}})(298\,\text{K})}\) Calculate the value of \(n\): \(n \approx 2.10\,\text{mol}\) Step 4: Calculate the new volume using the ideal gas law
04

Find the new volume

Given the new pressure, \(P' = 1.00\,\text{atm}\), and keeping the temperature constant at \(298\,\text{K}\), we can use the ideal gas law formula to find the new volume, \(V'\), of the Nâ‚‚O gas: \(V' = \frac{nRT'}{P'}\) Step 5: Plug in the values and calculate the new volume
05

Calculate the new volume

Substitute the values into the formula derived in Step 4: \(V' =\frac{(2.10\,\text{mol})(0.0821 \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}})(298\,\text{K})}{1.00\,\text{atm}}\) Calculate the value of \(V'\): \(V' \approx 513\,\text{L}\) So, there are approximately 2.10 moles of Nâ‚‚O gas in the cylinder, and if the pressure is reduced to 1.00 atm, the gas would occupy a volume of approximately 513 L.

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Key Concepts

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

Chemical Properties of Nitrous Oxide
Nitrous oxide (Nâ‚‚O), also known as laughing gas, is a colorless, non-flammable gas with a slightly sweet odor. It is an oxide of nitrogen and has several distinctive chemical properties that make it useful in various applications. Primarily, Nâ‚‚O supports combustion by releasing oxygen at high temperatures. This chemical property is beneficial in rocket motors and car racing. In the medical field, it is utilized for its anesthetic and analgesic effects and is considered safe as it does not react with most drugs and is excreted unchanged by the lungs. Nitrous oxide is also a powerful greenhouse gas with a global warming potential approximately 300 times that of carbon dioxide.

When studying nitrous oxide within the context of chemical reactions, it is important to understand its behavior under different temperatures and pressures. This leads us to the ideal gas law, a valuable tool in predicting the behavior of gases such as Nâ‚‚O under various conditions.
Application of Ideal Gas Law Formula
The Ideal Gas Law is a fundamental equation in the study of thermodynamics and is expressed as PV = nRT. Each symbol represents a crucial component: P for pressure, V for volume, n for the number of moles, R for the ideal gas constant, and T for temperature. This formula provides an accurate approximation for the behavior of many gases under normal conditions.

In application, you can manipulate the Ideal Gas Law to solve for any of the variables if the others are known. It becomes particularly useful in diverse fields such as chemistry, physics, engineering, and environmental science. For example, understanding the expansion and compression of gases is vital in engines, while calculating gas release in chemical reactions is essential for laboratory work and industrial processes.
Calculating Moles of Gas
Calculating the number of moles of a gas is a crucial step in chemical stoichiometry and involves using the Ideal Gas Law. The number of moles, n, can be found by rearranging the Ideal Gas Law equation to n = PV/RT. By dividing the product of pressure and volume by the product of the gas constant and temperature, you can determine how much of a gas is present in terms of moles.

Knowing the moles of a gas allows for further calculations, such as determining the mass of a gas using molar mass or calculating reactant and product quantities in a chemical reaction. Accurate mole calculations are crucial for lab work, as they can dictate the success of a reaction or the yield of a product.
Gas Law Calculations
Gas law calculations involve using the Ideal Gas Law and other related equations to determine various properties of gases. These calculations can provide valuable insights into how gases will respond to changes in conditions. For instance, when a gas is heated in a closed container, the pressure might increase, or the volume of a gas might expand if the pressure is decreased while the temperature is held constant.

To comprehensively understand gas law calculations, it's important to grasp the relationship between the variables. If temperature and pressure are constant, the volume of a gas is directly proportional to the number of moles. This understanding is crucial for tasks such as predicting how much space a certain mass of a gas will occupy under specific conditions, just like the expansion of Nâ‚‚O when it's transferred to a larger container at a lower pressure.

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Most popular questions from this chapter

Which of the following statements is(are) true? For the false statements, correct them. a. At constant temperature, the lighter the gas molecules, the faster the average velocity of the gas molecules. b. At constant temperature, the heavier the gas molecules, the larger the average kinetic energy of the gas molecules. c. A real gas behaves most ideally when the container volume is relatively large and the gas molecules are moving relatively quickly. d. As temperature increases, the effect of interparticle interactions on gas behavior is increased. e. At constant \(V\) and \(T\), as gas molecules are added into a container, the number of collisions per unit area increases resulting in a higher pressure. f. The kinetic molecular theory predicts that pressure is inversely proportional to temperature at constant volume and moles of gas.

The steel reaction vessel of a bomb calorimeter, which has a volume of \(75.0 \mathrm{~mL}\), is charged with oxygen gas to a pressure of 145 atm at \(22^{\circ} \mathrm{C}\). Calculate the moles of oxygen in the reaction vessel.

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