Question

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

There are approximately 2.10 moles of N₂O 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

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

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

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

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 N₂O gas: \(V' = \frac{nRT'}{P'}\) Step 5: Plug in the values and calculate the new volume

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 N₂O gas in the cylinder, and if the pressure is reduced to 1.00 atm, the gas would occupy a volume of approximately 513 L.

Key concepts

Chemical Properties of Nitrous Oxide

Nitrous oxide (N₂O), 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, N₂O 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 N₂O 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 N₂O when it's transferred to a larger container at a lower pressure.