Question

Propane, \(\mathrm{C}_{3} \mathrm{H}_{8},\) liquefies under modest pressure, allowing a large amount to be stored in a container. (a) Calculate the number of moles of propane gas in a \(110-\) L container at 3.00 atm and \(27^{\circ} \mathrm{C}\) (b) Calculate the number of moles of liquid propane that can be stored in the same volume if the density of the liquid is 0.590 \(\mathrm{g} / \mathrm{mL}\) . (c) Calculate the ratio of the number of moles of liquid to moles of gas. Discuss this ratio in light of the kinetic-molecular theory of gases.

Short answer

The number of moles of propane gas in a 110-L container at 3 atm and 27°C is approximately 14.6 moles. In its liquid state, occupying the same volume with a density of 0.590 g/mL, there are approximately 1470 moles of liquid propane. The ratio of moles between liquid and gas propane is approximately 100, which means there are 100 times more moles of liquid propane than gas propane in the same volume. This supports the kinetic-molecular theory of gases, as it demonstrates the large volume occupied by gas propane due to the empty space between its particles compared to the closely packed particles of liquid propane.

Step-by-step solution

Calculate the number of moles of propane gas

To calculate the number of moles of propane gas in gaseous state, we will use the Ideal Gas Law: \(PV = nRT\), 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 in Kelvin. Given: \(P = 3.00\,\text{atm}\) \(V = 110 \,\text{L}\) \(T = 27^\circ \text{C} = 300 \text{K}\) (Adding 273 to the given temperature in Celsius) Ideal gas constant, \(R = 0.0821\,\frac{\text{atm} \cdot \text{L}}{\text{mol} \cdot \text{K}}\) We need to find the value of \(n\). Rearrange the ideal gas equation for \(n\): \(n = \frac{PV}{RT}\)

Calculate the value of n for propane gas

Plug the given values into the equation: \(n = \frac{(3.00\,\text{atm})(110\,\text{L})}{(0.0821\,\frac{\text{atm} \cdot \text{L}}{\text{mol} \cdot \text{K}})(300\,\text{K})}\) \(n \approx 14.6\,\text{moles}\) So, there are approximately 14.6 moles of propane gas in the container.

Calculate the mass of liquid propane in the container

Given the density of liquid propane is: \(\rho = 0.590\,\frac{\text{g}}{\text{mL}}\) We also know the volume of the container is 110 L. Convert the volume from liters to milliliters: \(V = 110\,\text{L} \times \frac{1000\,\text{mL}}{\text{L}} = 110,000\,\text{mL}\) Now, we can calculate the mass of liquid propane using the formula: mass = density × volume mass = \((0.590\,\frac{\text{g}}{\text{mL}})(110,000\,\text{mL}) \approx 64900\,\text{g}\)

Calculate the number of moles of liquid propane

To find the number of moles of liquid propane, use the formula: \(n = \frac{\text{mass}}{\text{molar mass}}\) The molar mass of propane (C3H8) is 3(12.01) + 8(1.01) \(= 44.11\,\text{g/mol}\). Now to find the number of moles: \(n = \frac{64900\,\text{g}}{44.11\,\text{g/mol}} \approx 1470\,\text{moles}\) So, there are approximately 1470 moles of liquid propane in the container.

Calculate the ratio of moles between liquid and gas propane

To find the ratio of moles of liquid propane to gas propane, simply divide the number of moles of liquid propane by the number of moles of gas propane: Ratio = \(\frac{\text{moles of liquid propane}}{\text{moles of gas propane}}\) Ratio = \(\frac{1470\,\text{moles}}{14.6\,\text{moles}} \approx 100\)

Discuss the ratio in light of the kinetic-molecular theory of gases

According to the kinetic-molecular theory of gases, the particles in a gas are assumed to be far apart, and there is a lot of empty space between these particles. This assumption does not hold true for liquids, where the particles are much closer together. The ratio of 100 moles of liquid propane to 1 mole of gas propane supports this theory as it shows that gas propane takes up such a large volume compared to liquid propane because of the large amount of empty space between gas particles. Therefore, converting the propane from gas to the liquid state increases the number of available moles that can be stored in the same volume by a factor of 100.

Key concepts

Mole Calculations

Mole calculations are vital to converting between the mass of a substance and its corresponding number of particles. This process makes it easier to understand and analyze chemical quantities and reactions. When dealing with gases, the Ideal Gas Law equation, \(PV = nRT\), is often used for these calculations, as it connects pressure (\(P\)), volume (\(V\)), the number of moles (\(n\)), the ideal gas constant (\(R\)), and temperature (\(T\)). Let's break down these variables:

• Pressure (P): Measures the force exerted by gas particles against the walls of its container, specified here in atmospheres (atm).
• Volume (V): Represents the space the gas occupies, given in liters (L).
• Temperature (T): The measure of kinetic energy of the gas particles, converted to Kelvin by adding 273 to the Celsius temperature.
• Ideal Gas Constant (R): A fundamental constant (0.0821 atm·L/mol·K) that quantifies the relationship among the variables for an ideal gas.

Using these elements, you can calculate the number of moles of a gas if you know its pressure, volume, and temperature. By rearranging the formula, \(n = \frac{PV}{RT}\), you can easily substitute the known values to find the moles. This calculation was done for propane gas, resulting in approximately 14.6 moles of gas in the container, showcasing the utility of mole calculations in understanding chemical quantities.

Gas Density

Gas density is a concept that gives insights about how mass is distributed in a given volume of a gas. Densities of gases tend to be much lower compared to those of liquids and solids, which impacts how they're stored and utilized.To find the density, you need the mass of the gas distributed over its volume. However, in the context of comparing a gas with its liquid state, using density can shift the focus to storage capabilities. In the exercise at hand, the density of liquid propane is given as 0.590 grams per milliliter (g/mL), which allows us to calculate the mass when stored as a liquid in a 110-liter container. To make calculations cohesive, remember to convert liters to milliliters when dealing with densities traditionally given in g/mL.Understanding the density lets you calculate the number of moles in the liquid state. Here, the mass was found using mass = density \(\times\) volume, resulting in the mass as 64900 grams. Knowing the molar mass of propane is about 44.11 g/mol, you can subsequently find the moles of liquid propane by dividing mass by molar mass, yielding approximately 1470 moles.Thus, gas density not only helps grasp physical properties but also assists in determining how much of a substance can be stored under various conditions.

Kinetic-Molecular Theory

The kinetic-molecular theory provides a framework for understanding the behavior of gases. It posits that gas particles are in constant motion and occupy a large amount of space relative to their size. This space accounts for gases' compressibility and low density. One of the critical insights from kinetic-molecular theory is the vast difference between the molecular spacing in gases and liquids. In gases, particles are far apart, meaning there's a significant amount of unused space. On liquefaction, as seen with propane, these particles come closer together, massively increasing the number of moles that can fit in a given volume. This phenomenon is highlighted in the exercise by comparing the ratio between liquid and gas moles. The result showed a ratio of approximately 100. This indicates that, under the same conditions, liquid propane allows you to store vastly more moles compared to gaseous propane. The theory thus emphasizes:

• Gases' low density due to loosely packed particles.
• Increased density when converting to liquids due to closely packed particles.
• Understanding the drastic changes in storage efficiency for gases liquefied under moderate pressure.

Therefore, the kinetic-molecular theory helps explain why gases, like propane in gaseous versus liquid form, have such different storage capacities.