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

(a) 13.42 moles of propane gas (b) 1471.43 moles of liquid propane (c) The ratio of the number of moles of liquid propane to propane gas is 109.54. This large ratio indicates that, under modest pressure, the gas can be condensed into a liquid, which can be stored much more compactly and efficiently than in gas form. This is consistent with the kinetic-molecular theory, as the gas molecules are forced closer together under pressure, resulting in an increase in intermolecular forces and ultimately allowing the substance to turn into a liquid.

Step-by-step solution

Conversion of temperature to Kelvin

In order to use the ideal gas law, we must first convert the Celsius temperature to Kelvin. The formula is: \[ T (K) = T ( ^\circ C) + 273.15 \] For a temperature of 27°C: \[T (K) = 27 + 273.15 = 300.15 K\]

Calculate the number of moles of propane gas

We can use the ideal gas law equation \(PV = nRT\) to find the number of moles of propane gas (\(n\)). Plugging in the values, we have: \[3.00\,\text{atm} \times 110\,\text{L} = n \times 0.0821\, (\text{L}\cdot\text{atm/mol}\cdot\text{K}) \times 300.15\, \text{K}\] Now, solve for \(n\): \[n = \frac{3.00 \times 110}{0.0821 \times 300.15} = 13.42 \, \text{moles}\]

Calculate the mass of liquid propane

To calculate the mass of liquid propane in the container, we will use the density formula: \[mass = density \times volume\] Given the density of liquid propane as 0.590 g/mL, we first convert the container volume from liters to milliliters: \[110\, \text{L} \times \frac{1000\, \text{mL}}{1\, \text{L}} = 110,000\, \text{mL}\] Now, using the density formula, we have: \[mass = 0.590\, \text{g/mL} \times 110,000\, \text{mL} = 64,900\, \text{g}\]

Calculate the number of moles of liquid propane

Next, we will convert the mass of liquid propane to the number of moles using the molecular weight of propane (C3H8), which is: \[(3 \times 12.01\, \text{g/mol}) + (8 \times 1.01\, \text{g/mol}) = 44.1\, \text{g/mol}\] Now, divide the mass by the molecular weight to get the number of moles: \[n = \frac{64,900\, \text{g}}{44.1\, \text{g/mol}} = 1471.43\, \text{moles}\]

Calculate the ratio of the number of moles of liquid to moles of gas

Finally, we will determine the ratio of the number of moles of liquid propane to the number of moles of propane gas: \[\text{Ratio} = \frac{n_{liquid}}{n_{gas}} = \frac{1471.43}{13.42} = 109.54\] #Answer# (a) 13.42 moles of propane gas (b) 1471.43 moles of liquid propane (c) The ratio of the number of moles of liquid propane to propane gas is 109.54. This large ratio indicates that, under modest pressure, the gas can be condensed into a liquid, which can be stored much more compactly and efficiently than in gas form. This is consistent with the kinetic-molecular theory, as the gas molecules are forced closer together under pressure, resulting in an increase in intermolecular forces and ultimately allowing the substance to turn into a liquid.

Key concepts

Kinetic-Molecular Theory

The kinetic-molecular theory offers a way to understand the behavior of gases on a molecular level. It's a theory that explains gas properties in terms of the motion and energy of molecules. Here are the core ideas:

• Gases consist of large numbers of tiny particles that move randomly at high speeds.
• The size of the gas molecules compared to the distance between them is negligible, meaning most of the volume gas occupies is empty space.
• Collisions between gas particles and with the walls of containers are perfectly elastic, meaning there's no net loss of energy after collisions.
• The average kinetic energy of gas particles is directly proportional to the temperature of the gas in Kelvin. So, as temperature increases, kinetic energy and thus the speed of molecules also increase.

In the context of propane in our exercise, when we increase the pressure on the gas in a container, we push the molecules closer together, enhancing intermolecular interactions. As described in the kinetic-molecular theory, this interaction can result in the gas condensing into a liquid, as it becomes energetically favorable under certain conditions like lower temperature or higher pressure.

Propane Density

Density is how much mass is contained in a given volume. For a gas like propane, density can change significantly based on pressure and temperature. However, for liquids, the density is generally more consistent.

To find the density of liquid propane, the exercise mentions a given density of 0.590 g/mL. This tells us that every milliliter of liquid propane has 0.590 grams of mass.

This density allows us to calculate the mass of liquid propane that fits into a specific volume. For example, by knowing the volume in milliliters, we can multiply it by the density to determine the mass, as shown in the exercise.

This concept is essential, as it gives us insight into how much propane can fit inside a container when it is in its liquid form compared to when it is a gas. Typically, the liquid state allows us to store more propane in the same volume, making it efficient for storage and transportation.

Moles Calculation

The calculation of moles is a fundamental part of chemistry that involves converting between mass, number of particles, and volume of substances. The mole is a unit that helps bridge the atomic and macroscopic world.

In the context of the exercise, moles are calculated using different formulas depending on the state of propane (gas or liquid).

1. **For gases:** We use the ideal gas law equation, which is: \( PV = nRT \) where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the gas constant (0.0821 L·atm/mol·K),
• \( T \) is the temperature in Kelvin.

By rearranging the formula to solve for \( n \), we can find the number of moles of gas in a given volume and conditions.

2. **For liquids:** We start with the mass, calculated using the volume and the known density. Once we have the mass, we determine moles by dividing by the molecular weight of the substance. For propane, it's calculated based on its molecular formula \( \mathrm{C}_{3} \mathrm{H}_{8} \).

These calculations are crucial for understanding how much substance is present and are often used in chemical reactions to ensure the right proportions and amounts are involved.