Problem 119 - Chemistry: An Atoms-Focused Approach

Question

At high pressures, real gases do not behave ideally. a. Use the van der Waals equation and data in the text to calculate the pressure exerted by \(50.0 \mathrm{g}\) of \(\mathrm{H}_{2}\) at \(20^{\circ} \mathrm{C}\) in a \(1.00 \mathrm{L}\) container. b. Repeat the calculation assuming that the gas behaves like an ideal gas.

Step-by-Step Solution

Answer

Answer: To compare the pressures, first, convert the mass of hydrogen gas to moles and temperature to Kelvin. Then, calculate the pressures using both the van der Waals equation and the ideal gas law. Finally, compare the calculated pressures to see how the behavior of the real gas deviates from that of an ideal gas.

Step 1: Convert mass to moles and temperature to Kelvin

Firstly, we need to convert the given mass of hydrogen gas (50.0 g) to moles using the molar mass of H2. Also, we need to convert the given temperature (20°C) to Kelvin. Molar mass of H2 = 2 g/mol Number of moles (n) = mass / molar mass Temperature in Kelvin (T) = Temperature in °C + 273.15

Step 2: Calculate pressure using the van der Waals equation

To find the pressure using the van der Waals equation, we would need the van der Waals constants for H2, which can be found in the textbook. The equation is: P = [(nRT) / (V - nb)] - [a(n^2) / V^2] where P is the pressure, n is the number of moles, R is the gas constant (8.314 J/(K·mol)), T is the temperature in Kelvin, V is the volume, and a and b are the van der Waals constants. Insert the values and constants into the equation and solve for P.

Step 3: Calculate pressure assuming ideal gas behavior

Now, we will calculate the pressure exerted by the gas, assuming ideal gas behavior, using the ideal gas law: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (8.314 J/(K·mol)), and T is the temperature in Kelvin. Re-arrange the equation to find P: P = nRT / V Insert the values and constants into the equation and solve for P.

Step 4: Compare the pressures calculated using both equations

Now we have the pressures calculated using both the van der Waals equation and the ideal gas law, we can compare them to see how the behavior of the real gas differs from that of an ideal gas under high pressure conditions.

Key Concepts

Real Gases

When dealing with real gases, it's important to acknowledge that they don't always follow the simple laws defined for ideal gases. This deviation is especially noticeable at high pressures and low temperatures where the assumptions about gas molecules having negligible volume and no interactions don't hold true. Real gases have molecules with actual volume and forces acting between them.
To account for these real-world behaviors, we use the van der Waals equation. The equation adds two new parameters, 'a' and 'b', to correct for the intermolecular forces and the volume occupied by the gas molecules, respectively.

• 'a' accounts for the attractive forces between the particles. A higher 'a' value indicates stronger attraction.
• 'b' corrects for the volume occupied by gas molecules, so a larger 'b' suggests larger molecules.

These constants make the van der Waals equation crucial for accurately predicting real gas behavior and pressure in unique situations, like the one in our exercise.

Ideal Gas Law

The ideal gas law is a simplified model and is widely applicable under many conditions. It relates the pressure, volume, temperature, and amount of gas in moles in an equation: \[ PV = nRT \]where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the universal gas constant, and
• \( T \) is the temperature in Kelvin.

The ideal gas law assumes that the gas particles themselves occupy no volume and there are no intermolecular forces between them.
This means the gas behaves perfectly under low pressure and high temperature. Under these conditions, gases approximate the behavior described by the ideal gas law. However, there are limitations, especially when the gas is compressed or cooled, as in high-pressure environments.

Pressure Calculation

Pressure calculation for gases can be approached using different equations based on how closely reality follows the ideal conditions.

• In the real-world scenario with Hydrogen given in the problem, the pressure can be calculated with the van der Waals equation:\[ P = \left( \frac{nRT}{V - nb} \right) - \left( \frac{a(n^2)}{V^2} \right) \]Inserting values calculated earlier: the number of moles \( n \), the container volume \( V \), Temperature \( T \) and after obtaining constants \( a \) and \( b \) from tables, the pressure \( P \) can be determined.
• Assuming an ideal scenario, use the ideal gas law: \[ P = \frac{nRT}{V} \]This simplification often overestimates the pressure compared to the van der Waals equation under the same conditions due to lack of volume and attraction corrections.

Each method reveals different facets of gas behavior, and the resulting values show how assumptions affect calculations in practice.

Gas Behavior

Gas behavior can vastly differ based on environmental conditions and the nature of the gas itself.
At the molecular level, gases are composed of fast-moving particles that are spaced far apart. The behavior of these particles changes under different conditions.

• At high pressures, gas molecules are forced closer together, intensifying intermolecular forces. This leads to deviations from the ideal gas assumptions.
• At low temperatures, the reduced kinetic energy results in molecules interacting more strongly, further diverging from ideal predictions.

The real versus ideal gas behavior is notable during the calculation of pressure, as real gases don't always conform to neat mathematical predictions. This understanding is vital for interpreting results in fields ranging from industrial chemistry to atmospheric sciences. Appreciating these differences helps scientists and engineers accurately design experiments, processes, and devices involving gases.