Problem 103 - 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 \(3.00\) 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.

Step-by-Step Solution

Answer

In short, we first calculated the number of moles of propane gas in the container using the ideal gas law equation and obtained the result by plugging in the given values for pressure, volume, and temperature. Next, we calculated the mass of liquid propane that can be stored in the same container using the given density and then found the number of moles of liquid propane using the mass and molar mass. Finally, we found the ratio of the number of moles of liquid propane to the number of moles of gas propane and discussed how this ratio reflects the observations from the kinetic-molecular theory of gases, where liquid propane can be stored in higher amounts in the same volume compared to its gaseous state.

Step 1: Calculate the number of moles of propane gas

We will use the ideal gas law equation: PV=nRT Where: P = Pressure = 3.00 atm V = Volume = 110 L n = Number of moles (we need to find this) R = Ideal gas constant = 0.0821 \( \frac{\text{atm} \cdot \text{L}}{\text{mol} \cdot \text{K}} \) T = Temperature = 27ºC = 300 K We can rearrange the equation to find n: \( n = \frac{PV}{RT} \) Plugging in the values, we get: \( n = \frac{(3.00\text{ atm})(110\text{ L})}{(0.0821 \frac{\text{atm} \cdot \text{L}}{\text{mol} \cdot \text{K}})(300\text{ K})} \)

Step 2: Calculate the number of moles of liquid propane

Given the density of liquid propane as \(0.590 \frac{g}{mL}\), we can calculate the mass of the liquid propane that can be stored in the same 110-L container: Mass = density × volume = \( 0.590 \frac{g}{mL} \times 110000\,mL \) (convert L to mL) Now, we need to find the number of moles of liquid propane. The molar mass of propane, \(\mathrm{C}_{3} \mathrm{H}_{8}\), is 44.10 g/mol. We can now find the number of moles using the mass and molar mass: \( \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{\text{Mass of liquid propane}}{44.10} \)

Step 3: Calculate the ratio of the number of moles of liquid propane to the number of moles of gas propane

Now that we have the number of moles for both gas and liquid propane, we can calculate the ratio: \[ \text{Ratio} = \frac{\text{Number of moles of liquid propane}}{\text{Number of moles of gas propane}} \]

Step 4: Discuss the ratio in light of the kinetic-molecular theory of gases

The kinetic-molecular theory of gases states that gas particles are far apart from each other, and their volume can be neglected compared to the volume of the container. In the case of a liquid, the particles are close together, occupying a smaller volume than their gaseous counterparts. Since propane liquefies under modest pressure, it's expected that a greater number of moles of liquid propane can be stored in the same volume than in the gas state. The ratio calculated in Step 3 should correspond to these observations.

Key Concepts

Moles of Gas Calculation

Understanding how to calculate the moles of gas involves a fundamental piece of knowledge called the Ideal Gas Law. This law describes the relationship among pressure (P), volume (V), temperature (T), and the number of moles (n) of gas, expressed through the equation PV=nRT. In this equation, R represents the ideal gas constant.

The task for part (a) requires applying this equation to find out how many moles of propane gas are present in a known volume of space at a certain temperature and pressure. Just recall, you must first convert the temperature from degrees Celsius to Kelvin, as the Ideal Gas Law requires an absolute temperature scale. The conversion formula is K = °C + 273.15.

Once the ideal gas constant, temperature in Kelvin, pressure in atmospheres, and volume in liters are known, all you have to do is plug these values into the Ideal Gas Law and solve for n, the number of moles. In doing so, students can bridge the gap between abstract formulae and practical application, calculating precisely how much propane gas a container can hold.

Gas to Liquid Conversion

A fascinating aspect of substances like propane is their ability to exist in different states (gas and liquid) under varying conditions. To understand the transition from gas to liquid, called condensation, think of it as a density driven process. In part (b), a shift happens where propane changes from a less dense gas to a more dense liquid with the application of pressure.

For conversion, the key element is determining the mass of the liquid propane that the container can hold. This is obtained by multiplying the container's volume by the density of propane in its liquid form. However, to complete the understanding, you must recognize that density is mass per unit volume, and in this scenario, it's provided in grams per milliliter (g/mL).

After calculating the mass, students can utilize the molar mass of propane to find out the number of moles present in that mass. This calculation underscores the connectivity between different stages of matter, and quantifies how substance quantity remains consistent across these stages, even as their physical characteristics drastically change.

Kinetic-Molecular Theory of Gases

The Kinetic-Molecular Theory of Gases is a model that helps us grasp the behavior of gases and how it differs fundamentally from liquids. This theory espouses the idea that gas particles are in constant, random motion and that they are much farther apart compared to particles in a liquid. This explains why gases occupy more volume than a liquid would in the same container.

When examining the ratio in part (c), derived from the number of moles in liquid form versus gaseous form, it highlights this vast difference in volume occupation due to different particle arrangements. It is intriguing to observe how under modest pressure, propane can transition from a state with particles that are spread out to a state where particles are closely packed together, thus enabling a significantly higher number of moles of propane to be held in liquid form compared to gaseous form within the same volume.

This principle ties back to real-world applications, such as the storage of propane as a liquefied petroleum gas and points towards the efficiency of storing and transporting condensed forms of gaseous substances.