Problem 23 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A \(10 \mathrm{dm}^{3}\) SCUBA cylinder is filled with air to a pressure of \(\left.300 \text { atm at a temperature of } 20^{\circ} \mathrm{C}(293 \mathrm{K}) . \text { (Section } 8.6\right)\) (a) Calculate the amount in moles of gas in the cylinder, assuming the air behaves as an ideal gas. (b) When the diver jumps into cold water at \(278 \mathrm{K}\), the pressure gauge shows an alarming drop in pressure. Explain the reason why and calculate the new pressure inside the cylinder. (c) In fact, the compressed gases do not behave as ideal gases. Explain why. Use the van der Waals equation (for air, \(a=0.137 \mathrm{Pam}^{6} \mathrm{mol}^{-2}\) and \(b=3.7 \times 10^{-5} \mathrm{m}^{3} \mathrm{mol}^{-1}\) ) to show that the amount of air in the cylinder is 115 mol. In view of your answer to part (a) above, what are the implications of this for divers?

Step-by-Step Solution

Answer

Using the ideal gas law, 124.4 mol of air is in the cylinder at 20°C; pressure drops to 284.5 atm at 278 K. Real gases require corrections; actual amount shows deviation, affecting divers' air supply estimates.

Step 1: Convert Given Values to Required Units

The initial volume of the cylinder is given as 10 dm³, which we convert to m³ for standard volume in calculations: \[ 10 \text{ dm}^3 = 0.01 \text{ m}^3 \]The temperature of 20°C is converted to Kelvin for convenience in gas law formulas: \[ T_1 = 293 \text{ K} \] and the initial pressure is given as 300 atm.

Step 2: Use the Ideal Gas Law to Find Moles of Gas (Part a)

The ideal gas law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is moles, \( R \) is the universal gas constant \( 8.314 \text{ J/(mol·K)} \), and \( T \) is temperature.We solve for \( n \): \[ n = \frac{PV}{RT} \]Inserting the known values (convert pressure from atm to Pa: \( 300 \text{ atm} = 300 \times 101325 \text{ Pa} \)):\[ n = \frac{(300 \times 101325) \times 0.01}{8.314 \times 293} \]Calculate to find \( n \approx 124.4 \text{ mol} \).

Step 3: Calculate New Pressure Using Ideal Gas Law (Part b)

When the temperature drops to 278 K, we use the relation for ideal gases as \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \] where \( P_1 = 300 \text{ atm} \) and \( T_1 = 293 \text{ K} \).Rearrange to solve for \( P_2 \): \[ P_2 = P_1 \frac{T_2}{T_1} \]Substitute the known values:\[ P_2 = 300 \times \frac{278}{293} \] Calculate to find \( P_2 \approx 284.5 \text{ atm} \).

Step 4: Explain Deviation from Ideal Gas Behavior (Part c)

Real gases deviate from ideal behavior due to intermolecular forces and the finite size of molecules, which are not considered in the ideal gas law.The van der Waals equation accounts for these by adding two correction terms:\[ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \]Given values for air: \( a = 0.137 \text{ Pa m}^6 \text{ mol}^{-2} \), \( b = 3.7 \times 10^{-5} \text{ m}^3/\text{mol} \).Solve for \( n \) by rearranging the terms to confirm the amount of air is 115 mol, indicating non-ideal behavior.

Step 5: Consider Safety Implications for Divers

The deviation from ideal gas behavior means that calculations based solely on the ideal gas law might overestimate the amount of gas in the cylinder. For divers, this can be critical because relying on ideal gas estimates might lead to running out of breathable air sooner than expected. It is essential to consider real gas calculations or use correction factors for greater safety.

Key Concepts

Van der Waals equation

In the world of chemistry, the Van der Waals equation is a critical tool used to understand how real gases deviate from ideal behavior. Real gases do not always follow the Ideal Gas Law due to the interactions that occur between gas molecules and the volume they occupy. This is where the Van der Waals equation comes into play.

It is written as: \[\left(P + \frac{an^2}{V^2}\right)(V-nb) = nRT \] where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of the gas.
• \(R\) is the universal gas constant (8.314 J/(mol·K)).
• \(T\) is the absolute temperature in Kelvin.
• \(a\) and \(b\) are specific constants for each gas, accounting for intermolecular forces and molecular volume, respectively.

The equation modifies the Ideal Gas Law by introducing correction factors:

• The term \(\frac{an^2}{V^2}\) corrects for the attraction between molecules.
• The term \(nb\) adjusts for the volume occupied by the gas molecules themselves.

When using these adjustments, we can determine more accurately the conditions of real gases, such as when they're compressed in cylinders, ensuring greater accuracy over the Ideal Gas Law.

Real gases vs. Ideal gases

Ideal gases are a theoretical concept wherein gas molecules are assumed to have no volume and no interaction forces between them. These assumptions help simplify calculations, making it easier to predict gas behavior using the Ideal Gas Law. However, real gases exist in the world, and they do have volume and experience intermolecular forces.

The limitations of the Ideal Gas Law become evident under conditions of high pressure and low temperature. Here's why:

• **High Pressure:** At high pressures, the volume of gas particles is no longer negligible compared to the volume of the container.
• **Low Temperature:** As temperatures drop, molecules move slower, and intermolecular forces like van der Waals forces become more significant.

Real gases deviate from ideal behavior due to these very molecular volumes and interactions. The Van der Waals equation provides a refined approach that accounts for these deviations, ensuring a more accurate representation of gas behavior in real-world conditions.

Temperature and pressure relationship

Temperature and pressure have a direct relationship in gases, often described by Charles' and Gay-Lussac's Law. This is deeply rooted in the kinetic molecular theory which states that as temperature increases, gas particles move more quickly, impacting pressure.

For ideal gases, at a constant volume, the law can be expressed as:\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]where \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature, respectively.

When a diver's cylinder is submerged into colder water, the temperature of the gas inside the cylinder decreases. Since volume stays constant, this results in a drop in pressure, which is exactly what is observed when the pressure gauge reads lower. Understanding this relationship is crucial, especially in practical applications like SCUBA diving. Incorrect assumptions about pressure could lead to inaccurate calculations of gas supply, impacting a diver's safety.