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Chapter 10: Problem 53

In the text, it is stated that the pressure of 4.00 mol of \(\mathrm{Cl}_{2}\) in a \(4.00-\mathrm{L}\) tank at \(100.0^{\circ} \mathrm{C}\) should be 26.0 atm if calculated using the van der Waals equation. Verify this result, and compare it with the pressure predicted by the ideal gas law.

Short Answer

Expert verified
The van der Waals pressure is 26.0 atm; the ideal gas law gives 30.66 atm.

Step by step solution

01

Understand the Van der Waals Equation

The van der Waals equation is given by \[\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\] where \(P\) is the pressure, \(n\) is the number of moles, \(V\) is the volume, \(T\) is the temperature in Kelvin, \(R\) is the ideal gas constant \((0.0821 \, \text{L atm/mol K})\), and \(a\) and \(b\) are specific constants for chlorine gas: \(a = 6.49 \, \text{L}^2\text{atm/mol}^2\), \(b = 0.0562 \, \text{L/mol}\).
02

Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin. \[T = 100.0 + 273.15 = 373.15 \, \text{K}\]
03

Substitute Values into Van der Waals Equation

Substitute all known values into the van der Waals equation: \[\left(P + \frac{6.49 \times 4.00^2}{4.00^2}\right)(4.00 - 4.00 \times 0.0562) = 4.00 \times 0.0821 \times 373.15\] Simplifying this gives \(\left(P + 6.49\right)(3.7752) = 122.6312\)
04

Solve for Pressure P

Solve the equation for \(P\): \[\left(P + 6.49\right) = \frac{122.6312}{3.7752} = 32.4784\] \[P = 32.4784 - 6.49 = 25.99 \, \text{atm}\] The calculated pressure is approximately 26.0 atm.
05

Use the Ideal Gas Law

The ideal gas law is \(PV = nRT\). Substitute the known values: \[P \times 4.00 = 4.00 \times 0.0821 \times 373.15\] \[P = \frac{122.6312}{4.00} = 30.66 \, \text{atm}\]
06

Compare Results

The pressure from the van der Waals equation is 26.0 atm, while the ideal gas law predicts 30.66 atm. The non-ideal behavior described by van der Waals gives a lower pressure than predicted by the ideal gas law.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law
The ideal gas law is a simple yet powerful relation given by the equation \(PV = nRT\), where:
  • \(P\) is the pressure of the gas.
  • \(V\) is the volume the gas occupies.
  • \(n\) represents the number of moles of the gas.
  • \(R\) is the ideal gas constant (0.0821 L atm/mol K).
  • \(T\) is the temperature measured in Kelvin.
This equation assumes that gases consist of many small particles moving randomly and that they exert force during collisions with the walls of their container.
The assumptions also include that these particles do not interact with each other except for collisions.
The simplistic nature of this law makes it very useful for calculations where the gas behaves ideally, meaning the gas particles are assumed to have no volume and no interactions.
Chlorine Gas Constants
When dealing with chlorine gas, constants play a significant role in making precise calculations.
In the van der Waals equation, two constants are used for chlorine:
  • \(a = 6.49 \, \text{L}^2\text{atm/mol}^2\)
  • \(b = 0.0562 \, \text{L/mol}\)
The constant \(a\) corrects for the intermolecular forces present in chlorine gas.
In non-ideal gases, these forces cause the pressure to deviate from what is predicted by the ideal gas law.
The constant \(b\) accounts for the finite size occupied by chlorine molecules, further affecting the volume calculations.
Pressure Calculation
Pressure is calculated differently depending on the assumptions made about the gas.
For ideal gases, the pressure is calculated using the ideal gas law, where no forces act between the gas molecules.
In this exercise, the ideal gas law predicted a pressure of 30.66 atm for the chlorine gas.
However, using the van der Waals equation, we see a different picture due to the factors of non-ideal behavior.
Here, pressure is adjusted by incorporating the impact of intermolecular forces and the physical size of the molecules with their constants \(a\) and \(b\).
This results in a pressure of 26.0 atm, which is more accurate for the non-ideal behavior of chlorine gas.
Non-Ideal Gas Behavior
Non-ideal gas behavior is observed when gases deviate from the assumptions of the ideal gas law.
In real situations, gas molecules attract each other due to intermolecular forces, and they occupy space as they have volume.
This is particularly significant at high pressures and low temperatures, where the particles are closer together, enhancing these deviations.
The van der Waals equation is an adjustment to the ideal gas law that compensates for these factors by using corrected pressure and volume terms.
This allows for more accurate predictions and calculations, particularly for gases like chlorine that do not behave ideally under the given conditions.

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