Question

\(\mathrm{N}_{2} \mathrm{O}\) is a gas commonly used to help sedate patients in medicine and dentistry due to its mild anesthetic and analgesic properties; it is also nonflammable. If a cylinder of \(\mathrm{N}_{2} \mathrm{O}\) is at 10.5 atm and has a volume of 5.00 \(\mathrm{L}\) at \(298 \mathrm{K},\) how many moles of \(\mathrm{N}_{2} \mathrm{O}\) gas are present? The gas from the cylinder is emptied into a large balloon at 745 torr. What is the volume of the balloon at 298 \(\mathrm{K}\) ?

Short answer

The number of moles of N₂O gas present in the cylinder is approximately 2.13 moles. After the gas is emptied into a large balloon at 745 torr and 298 K, the volume of the balloon is approximately 53.2 L.

Step-by-step solution

Identify the given information for the first problem

The cylinder contains N₂O gas at a pressure of 10.5 atm, a volume of 5.00 L, and a temperature of 298 K. We will use these values along with the ideal gas law constant R=0.0821 L.atm/mol.K to find the number of moles (n) of N₂O gas present.

Apply the Ideal Gas Law formula for the first problem

The ideal gas law formula is PV=nRT. We know the values for P, V and T, we will solve for the number of moles (n). \( n = \frac{P * V}{R * T} \)

Calculate the number of moles of N₂O gas for the first problem

Substitute the known values into the formula: \( n = \frac{(10.5\: atm) * (5.00\: L)}{(0.0821\: L.atm/mol.K) * (298\: K)} \) After the calculation, we get: \( n \approx 2.13\: moles \) So, there are approximately 2.13 moles of N₂O gas present in the cylinder. Now, we will proceed to the second part of the problem.

Identify the given information for the second problem

The gas is emptied into a balloon at 745 torr and has a pressure of 745 torr which is approximately equal to 0.979 atm (1 atm = 760 torr). Temperature remains at 298 K, and we have already found the number of moles of N₂O gas, which is 2.13 moles.

Apply the Ideal Gas Law formula for the second problem

We will use the ideal gas law formula again, this time to find the balloon's volume (V). \( V = \frac{n * R * T}{P} \)

Calculate the volume of the balloon for the second problem

Substitute the known values into the formula: \( V = \frac{(2.13\: moles) * (0.0821\: L.atm/mol.K) * (298\: K)}{0.979\: atm} \) After the calculation, we get: \( V \approx 53.2\: L \) The volume of the balloon at 298 K when filled with the N₂O gas is approximately 53.2 L.

Key concepts

Moles of Gas

Understanding the concept of moles is crucial when working with gases and chemical reactions. The mole is a fundamental unit in chemistry that provides a way to quantify the amount of substance. When we calculate moles in the context of gases, we use the Ideal Gas Law. The formula is given as \( PV = nRT \), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume.
• \(n\) represents the number of moles.
• \(R\) is the ideal gas constant with a value of \(0.0821\, \text{L.atm/mol.K}\).
• \(T\) is the temperature in Kelvin.

To find the moles \(n\), the formula rearranges to \( n = \frac{PV}{RT} \). By plugging the given pressure, volume, and temperature into this equation, we can solve for the number of moles. In the exercise, we found approximately 2.13 moles of \(\text{N}_2\text{O}\) gas were present in the cylinder. This represents the quantity of gas particles in the cylinder at the given conditions.

Pressure Conversion

Converting pressure units is an important step in many gas law problems. In this exercise, we dealt with two different units: atm and torr. To apply formulas like the Ideal Gas Law, it is necessary to have consistent units.1 atm is equivalent to 760 torr. Knowing this, you can easily convert between these units using the conversion factor. In the second part of our problem:

• We needed to convert 745 torr to atm.
• This is done by dividing 745 by 760, which gives \(\approx 0.979\, \text{atm}\).

Converting the pressure to atm allows us to use the Ideal Gas Law effectively. Keeping track of unit conversions is critical in accurately solving gas law problems. Always ensure pressure, volume, and temperature are in the right units before solving.

Calculation of Volume

Finding the volume of a gas requires knowing its moles, pressure, and temperature, which are part of the Ideal Gas Law. For the exercise, with a known amount of gas from previously calculated moles, and the new conditions of pressure and temperature, we examined the final volume.The Ideal Gas Law can also be rearranged to solve for volume \(V\):\[ V = \frac{nRT}{P} \]Let's break this down:

• Use the number of moles \(n\) calculated from the initial condition.
• Keep the constant \(R\) as \(0.0821\, \text{L.atm/mol.K}\).
• Apply the temperature in Kelvin, which was \(298\, \text{K}\) in this problem.
• And the converted pressure \(P\) from torr to atm, \(0.979\, \text{atm}\).

By substituting these values into our equation, we find the new volume of the balloon was approximately \(53.2\, \text{L}\). This gives us the full capacity needed for the gases based on their new conditions, highlighting the dynamic behavior of gases under different conditions.