Problem 95 - Chemistry: The Central Science

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 \(3.00 \mathrm{~atm}\) 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 kineticmolecular theory of gases.

Step-by-Step Solution

Answer

In short, the number of moles of gaseous propane in a 110-L container at 3.00 atm and 27°C is calculated using the ideal gas law formula, \( n=\frac{PV}{RT}\), while the number of moles of liquid propane in the same volume is found using the mass calculated from density and molar mass. The ratio of moles of liquid to moles of gas indicates deviation from ideal gas behavior, with a higher ratio meaning a greater deviation. According to the kinetic molecular theory of gases, this can be explained by the greater attractive forces and volume of particles in liquid propane compared to an ideal gas.

Step 1: Calculate the number of moles of propane gas using the ideal gas law formula

To calculate the number of moles of the propane gas, we will use the ideal gas law formula \(PV = nRT\). We have the volume V = 110 L, and the pressure P = 3.00 atm. The temperature should be converted to Kelvin: \(T = 27^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{~K}\). The gas constant value for the ideal gas law in atm L is \(R = 0.0821 \frac{\mathrm{L }\cdot \mathrm{atm}}{\mathrm{mol }\cdot \mathrm{K}}\). By rearranging the formula, we can calculate the number of moles: \[n = \frac{PV}{RT}\]

Step 2: Calculate the number of moles of propane in its liquid form

First, we will convert the volume of the container from liters to milliliters to match the density units: \[110 \mathrm{~L} = 110,000 \mathrm{~mL}\] Now we will find the mass of the liquid propane by multiplying the volume by its density: \[m = V \cdot \rho = 110,000 \mathrm{~mL} \cdot 0.590 \frac{\mathrm{g}}{\mathrm{mL}} = 64,900 \mathrm{~g}\] Next, we will calculate the number of moles in the liquid propane by dividing the mass by the molar mass of propane. The molar mass of propane is calculated by finding the sum of the three carbon atoms and eight hydrogen atoms: \[M_\mathrm{propane} = 3M_\mathrm{C} + 8M_\mathrm{H} = 3(12.01 \frac{\mathrm{g}}{\mathrm{mol}}) + 8(1.01 \frac{\mathrm{g}}{\mathrm{mol}}) = 44.11 \frac{\mathrm{g}}{\mathrm{mol}}\] Then, the number of moles of liquid propane can be found using: \[n_\mathrm{liquid} = \frac{m}{M_\mathrm{propane}}\]

Step 3: Calculate the ratio of the number of moles of liquid to moles of gas and discuss it in light of the kinetic molecular theory of gases

To find the ratio of the number of moles of liquid to moles of gas, we will divide the number of moles of liquid propane by the number of moles of gaseous propane: \[\text{Ratio} = \frac{n_\mathrm{liquid}}{n_\mathrm{gas}}\] We will discuss this ratio in light of the kinetic molecular theory of gases. The kinetic molecular theory of gases states that the particles of an ideal gas are in constant motion, and they exert negligible volume and negligible attractive forces. The more the number of moles of the liquid, the greater the attractive forces and volume of particles compared to an ideal gas. Thus, the higher the ratio, the more deviation there is from ideal gas behavior for the liquid propane.

Key Concepts

Moles Calculation

Understanding how to calculate the number of moles of a substance is essential in chemistry. A mole is a unit of measurement that represents a set number of particles, atoms, ions, or molecules. It is equivalent to Avogadro's number, which is approximately 6.022 x 10^23 entities. To calculate the number of moles (), one can use the mass of the sample divided by the molar mass of the substance. In the case of propane (C_3H_8), one would divide the mass of propane by its molar mass (44.11 g/mol).

The mass can be found in different states of matter, such as in the gaseous form, where the ideal gas law is utilized, or in the liquid form, where the density is taken into account. The formula for moles calculation when given mass and molar mass is: \[n = \frac{m}{M}\]where m is the mass in grams and M is the molar mass. Calculating moles helps us quantify chemical reactions and understand the stoichiometry of compounds, providing the basis for further calculations in chemistry.

Kinetic Molecular Theory

The kinetic molecular theory (KMT) is a cornerstone of gas laws and explains the behavior of gas particles. This theory posits that gas particles are in constant, random motion and that they collide with each other and with the walls of their container. These collisions are elastic, meaning there is no net loss of energy from the collisions. KMT also assumes that gas particles are so small compared to the distances between them that their volume is negligible. Moreover, no forces of attraction or repulsion exist between the particles.

When discussing the moles of a gas versus a liquid, like in the case of propane, the increased number of moles in the liquid state indicates a significant deviation from the KMT. This is due to the liquid particles being closer together, thus exerting a volume and attractive forces that are significant, in contrast to their gaseous counterparts. Therefore, the ratio of moles in the liquid phase to the gaseous phase sheds light on the differences in particle behavior between the two states.

Propane Properties

Propane (C_3H_8) is a hydrocarbon that is widely used as a fuel. It exists in both gaseous and liquid states under common conditions. One of the most interesting properties of propane is its ability to be liquefied under modest pressure, which allows for efficient storage and transport. In its gaseous form, propane is colorless and odorless, although an odorant is usually added for safety purposes.

The physical properties of propane, such as density, boiling point, and molar mass, play a pivotal role in how it behaves and is manipulated in industrial and domestic settings. The density of liquid propane is about 0.590 g/mL, which is significantly more dense than its gaseous form and explains why a greater amount of propane can be stored as a liquid. Understanding the properties of propane, particularly how it can be converted between its liquid and gaseous states, is crucial for anyone working with this common fuel source.