Problem 102 - Chemistry The Central Science In Si Units

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 20 -L container at \(709.3 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C} .\) (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 the 20 L container, 5.71 moles of gas and 267.57 moles of liquid propane can be stored. The liquid to gas mole ratio is approximately 46.89.

Step 1: Convert Pressure and Temperature for Gas Calculation

The pressure given is in kilopascals (kPa) and should be converted to atmospheres (atm) for using the ideal gas law. Use the conversion: \(1 \text{ atm} = 101.3 \text{ kPa}\). \[ \text{Pressure in atm} = \frac{709.3 \text{ kPa}}{101.3 \text{ kPa/atm}} \approx 7.00 \text{ atm}\] The temperature in degrees Celsius is converted to Kelvin: \[ T = 25 + 273.15 = 298.15 \text{ K} \]

Step 2: Use Ideal Gas Law to Find Moles of Gas

We use the ideal gas law \( PV = nRT \) to find the moles of gas \(n\), where \(P\) is pressure, \(V\) is volume, \(R\) is the gas constant (\(0.0821 \text{ L atm/mol K}\)), and \(T\) is temperature.Plug in the values: \[ n = \frac{PV}{RT} = \frac{(7.00 \text{ atm})(20 \text{ L})}{(0.0821 \text{ L atm/mol K})(298.15 \text{ K})} \approx 5.71 \text{ moles} \]

Step 3: Calculate Moles of Liquid Propane

The density of liquid propane is given as \(0.590 \text{ g/mL}\). Convert the volume to milliliters: \[ 20 \text{ L} = 20000 \text{ mL} \]Then calculate the mass of the propane: \[ \text{Mass} = 0.590 \text{ g/mL} \times 20000 \text{ mL} = 11800 \text{ g} \]Next, use the molar mass of propane (\(\text{C}_{3}\text{H}_{8}\), approximately \(44.1 \text{ g/mol}\)): \[ n = \frac{11800 \text{ g}}{44.1 \text{ g/mol}} \approx 267.57 \text{ moles} \]

Step 4: Calculate the Ratio of Liquid to Gas Moles

Calculate the ratio of the moles of liquid propane to moles of gas propane found previously: \[ \text{Ratio} = \frac{267.57 \text{ moles}}{5.71 \text{ moles}} \approx 46.89 \]

Step 5: Analyze the Ratio in Context of Kinetic-Molecular Theory

The kinetic-molecular theory explains that gases have widely spaced molecules and occupy larger volumes than liquids. Since liquid propane is denser, it occupies less volume, allowing significantly more moles of propane to be stored. This explains the high ratio of liquid to gas moles.

Key Concepts

Kinetic-Molecular Theory

The kinetic-molecular theory provides insight into the behavior of gases. It explains how and why gases behave in certain ways under different conditions.
This theory states that gas molecules are in continuous, random motion, and this motion is what causes pressure when the molecules collide with the walls of their container.
Moreover, gas molecules are much farther apart compared to those in liquids, which results in them occupying a larger volume.

In contrast, the molecules in a liquid are closely packed due to stronger intermolecular forces. This compact arrangement allows a small volume to contain a large number of molecules.
This is clearly observed when comparing the number of moles in the same volume of liquid propane versus gaseous propane.

• The liquid can store more than 46 times the moles of gas in the same space.
• This difference supports the kinetic-molecular theory, illustrating why liquid storage is more efficient.

Density of Liquids

Density is a crucial property that determines how much mass is contained in a given volume.
It is calculated as mass per unit volume, typically expressed in grams per milliliter (g/mL) for liquids.

In our exercise, the liquid propane has a density of 0.590 g/mL.
To store propane efficiently, its liquid form is preferred due to its higher density compared to its gaseous state.

This results in more propane being stored in the same volume when liquefied.

• The 20-L container can hold a significant mass due to this density.
• Thus, density becomes a key factor when storing gases in liquid form for high efficiency.

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases, relating their pressure, volume, temperature, and amount.
The ideal gas law, expressed as \( PV = nRT \), is particularly useful for calculations involving gases.

In this ideal gas equation:

• \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) is the gas constant, and \(T\) represents temperature in Kelvin.
• It helps calculate the number of moles when pressure, volume, and temperature are known, as it was used to find the moles of propane gas in the exercise.

Importantly, gas laws allow for predictions regarding gas behavior under different scenarios.

Understanding them is critical when designing processes for gas storage and usage.

Molar Mass Calculation

Molar mass is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol).
It is calculated by summing the atomic masses of all atoms in a molecule.

For example, the molar mass of propane (\( \text{C}_3\text{H}_8 \)) is calculated as:

• Three carbon atoms: \(3 \times 12.01 \approx 36.03 \text{ g/mol}\)
• Eight hydrogen atoms: \(8 \times 1.008 \approx 8.064 \text{ g/mol}\)
• Total: \(44.1 \text{ g/mol}\)

This value is crucial for converting between mass and moles in chemical calculations.
In the provided exercise, it was used to determine the number of moles of liquid propane in a container by dividing the total mass of propane by its molar mass.

Understanding molar mass connects the microscopic world of molecules to macroscopic observations in chemistry.