Question

You have two rigid gas cylinders. Gas cylinder A has a volume of \(48.2 \mathrm{~L}\) and contains \(\mathrm{N}_{2}(g)\) at 8.35 atm at 25 . \(\mathrm{C}\). Gas cylinder \(\mathrm{B}\) has a volume of \(22.0 \mathrm{~L}\) and contains \(\mathrm{He}(g)\) at \(25 \quad \mathrm{C}\). When the two cylinders are connected with a valve of negligible volume and the gases are mixed, the pressure in each cylinder becomes 8.71 atm. (Assume no reaction when the gases are mixed.) a. How many nitrogen molecules are present? b. What is the total number of moles of \(\mathrm{N}_{2}(g)\) and \(\mathrm{He}(g)\) present after the gases are mixed? c. What was the beginning pressure of cylinder B containing only the \(\mathrm{He}(g)\) (i.e., before the valve was connected)? d. Think about the \(\operatorname{He}(g)\) before and after the cylinders were connected. Graph the relationship between pressure and volume (without numbers) for the \(\mathrm{He}(g)\) showing this change, and explain your answer, making sure to address the variables \(P, V, n,\) and \(T\)

Short answer

a. There are initially \(9.81×10^{24}\) nitrogen molecules in cylinder A. b. The total number of moles of N2 and He present after the gases are mixed is 32.2 moles. c. The beginning pressure of cylinder B containing only the He gas was 4.68 atm. d. The graph should show an upward curve connecting the initial state (P=4.68 atm, V=22.0 L) and the final state (P=8.71 atm, V=70.2 L), resembling the shape of a typical Boyle's Law graph. This demonstrates the inverse relationship between pressure and volume while holding the number of moles and temperature constant.

Step-by-step solution

List the given values and the Ideal Gas Law formula

The given values are: - Volume of cylinder A: \(V_A = 48.2 L\) - Pressure of cylinder A: \(P_A = 8.35 atm\) - Temperature of both cylinders: \(T = 25°C = 298 K\) (converted to Kelvin) - Volume of cylinder B: \(V_B = 22.0 L\) The Ideal Gas Law formula is: \(PV=nRT\) where: P = pressure in atm V = volume in L n = number of moles of gas R = gas constant = 0.0821 \(L.atm.K^{-1}.mol^{-1}\) T = temperature in K

Calculate the initial number of moles of nitrogen gas in cylinder A

To find the initial number of moles of nitrogen in cylinder A, we will use the Ideal Gas Law formula and rearrange it to solve for n: \(n=\frac{PV}{RT}\) Using the values for cylinder A: n = \(\frac{8.35 atm × 48.2 L}{0.0821 L.atm.K^{-1}.mol^{-1}× 298 K}\) n = 16.3 moles So there are initially 16.3 moles of nitrogen gas in cylinder A. a. Since each mole of nitrogen has \(6.022×10^{23}\) molecules (Avogadro's number), we can find the number of nitrogen molecules: Number of nitrogen molecules = 16.3 moles × \(6.022×10^{23}\) molecules/mole = \(9.81×10^{24}\) molecules

Calculate the total volume and use it to find the total pressure after mixing

Calculate the total volume when both cylinders are connected: Total volume = \(V_A + V_B = 48.2 L + 22.0 L = 70.2 L\) We are given the final pressure after the gases are mixed (8.71 atm). Hence, we can find the total moles of N2 and He gas after mixing using the Ideal Gas Law: \(n_{total}=\frac{P_{total}V_{total}}{RT}\) \(n_{total} = \frac{8.71 atm × 70.2 L}{0.0821 L.atm.K^{-1}.mol^{-1} × 298 K}\) \(n_{total} = 32.2\) moles b. The total number of moles of N2 and He present after the gases are mixed is 32.2 moles.

Calculate the initial number of moles of helium gas in cylinder B

To find the initial number of moles of helium gas in cylinder B, the difference between the total moles of gas after mixing and the initial moles of nitrogen gas in cylinder A can be calculated: Initial moles of helium = \(n_{total} - n_{N2} = 32.2 moles - 16.3 moles = 15.9\) moles

Calculate the initial pressure of helium gas in cylinder B

Now that we know the initial moles of helium gas in cylinder B, we can use the Ideal Gas Law to find the initial pressure of helium in cylinder B: \(P_{He} =\frac{n_{He}RT}{V_B}\) \(P_{He} = \frac{15.9 moles × 0.0821 L.atm.K^{-1}.mol^{-1} × 298 K}{22.0 L}\) \(P_{He} = 4.68 atm\) c. The beginning pressure of cylinder B containing only the He gas was 4.68 atm.

Graph the relationship between pressure and volume for He gas and explain

d. To graph the relationship between pressure and volume for the helium gas, we need to consider the gas's initial and final states before and after the cylinders are connected. Before connecting the cylinders, helium gas is confined to a smaller volume (22.0 L) and a lower pressure (4.68 atm). After connecting the cylinders, the volume increases to 70.2 L (total volume), whereas the final pressure is constant (8.71 atm) which is higher than the initial pressure of cylinder B. In this process, the number of moles of gas (n) and the temperature (T) remain constant. The graph should show a curve connecting the initial state (P=4.68 atm, V=22.0 L) and the final state (P=8.71 atm, V=70.2 L). The upward curve demonstrates the relationship between pressure and volume, which is inversely proportional, due to holding the number of moles and temperature constant. The graph should resemble the shape of a typical Boyle's Law graph, which states: \(P_1V_1=P_2V_2\) for a gas with constant n and T.

Key concepts

Gas Pressure

Gas pressure is crucial to understand when studying the behavior of gases. It's defined as the force applied by gas molecules as they collide with the surfaces of their container. The standard unit for measuring gas pressure is the atmosphere (atm), although other units like the Pascal (Pa) or pounds per square inch (psi) are also used. Students often encounter exercises where they must calculate the pressure of a gas within a container using the Ideal Gas Law formula, which relates pressure (P) with volume (V), number of moles (n), and temperature (T).

When a gas cylinder's valve is opened and the gas is allowed to mix into a second cylinder, the pressure within each cylinder can change. Understanding how to use the Ideal Gas Law to find this new pressure is a crucial skill in chemistry. For instance, if we know the volume and temperature of the gas, and the amount of substance present, we can find the pressure by re-arranging the Ideal Gas Law, showing that these variables are directly related to each other.

Gas Volume

Gas volume is another key concept within the context of the Ideal Gas Law. It represents the amount of space that a gas occupies and is typically measured in liters (L) or cubic meters (m^3). Volumes of gases are highly variable and can change with pressure and temperature.

In problems like the one about the gas cylinders, knowing the volume of each container and the combined volume when they are connected is essential for solving for the unknowns. With the Ideal Gas Law, students can calculate how volume affects other properties of the gas, like pressure, when the number of moles and temperature stays constant. For example, if the initial volumes of two gases before mixing are given, and we know the pressure and temperature, we can predict the new condition after they have combined, showing the direct proportionality between volume and the amount of gas when pressure and temperature are held steady.

Molar Volume

The molar volume of a gas relates to the volume that one mole of a gas occupies under specified conditions, typically at standard temperature and pressure (STP). At STP (0°C and 1 atm), one mole of an ideal gas occupies 22.4 L, which is known as the molar volume of an ideal gas at STP. However, this value changes with temperature and pressure, and calculations often require adjusting to the situation at hand.

Using molar volume is important when calculating the number of moles during gas mixture exercises like the cylinder example. With the combined volume known and assuming ideal conditions, you can determine the total number of moles using the Ideal Gas Law. This helps you to understand the proportion of different gases involved and their individual contributions to the pressure of the gas mixture.

Boyle's Law

Boyle's Law is a fundamental principle in understanding gas behaviors and is part of the greater Ideal Gas Law. It states that for a given amount of gas at constant temperature (isothermal conditions), the product of the pressure (P) and volume (V) of the gas is constant. This means that pressure and volume are inversely related; as one increases, the other decreases.

In our scenario with the gas cylinders, when the helium gas from cylinder B is allowed to expand into the larger combined volume of cylinders A and B, Boyle's Law explains this process. A visual depiction, such as a graph of pressure versus volume, can illustrate how an increase in volume leads to a decrease in pressure, and vice versa, keeping the number of moles and temperature constant. This graphical representation allows students to conceptualize the Boyle's Law relationship, giving a visual framework to the abstract concepts of gas properties.