Textbook Question
The graph shows the distribution of molecular velocities for the same molecule at two different temperatures (T1 and T2). Which temperature is greater? Explain.
Ch.6 - Gases
Chapter 6, Problem 93
Use the van der Waals equation and the ideal gas equation to calculate the volume of 1.000 mol of neon at a pressure of 500.0 atm and a temperature of 355.0 K. Explain why the two values are different. (Hint: One way to solve the van der Waals equation for V is to use successive approximations. Use the ideal gas law to get a preliminary estimate for V.)
Verified step by step guidance1
Start by using the ideal gas law, which is given by the equation: \( PV = nRT \). Here, \( P = 500.0 \text{ atm} \), \( n = 1.000 \text{ mol} \), \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \), and \( T = 355.0 \text{ K} \). Solve for \( V \) to get an initial estimate.
Calculate the initial volume estimate using the ideal gas law: \( V = \frac{nRT}{P} \). Substitute the known values into the equation to find \( V \).
Next, use the van der Waals equation: \[ \left( P + \frac{an^2}{V^2} \right)(V - nb) = nRT \]. For neon, the van der Waals constants are \( a = 0.211 \text{ L}^2\text{ atm mol}^{-2} \) and \( b = 0.0171 \text{ L mol}^{-1} \).
Substitute the initial volume estimate from the ideal gas law into the van der Waals equation to solve for \( V \) using successive approximations. Adjust \( V \) iteratively until the equation is satisfied.
Compare the volumes obtained from the ideal gas law and the van der Waals equation. The difference arises because the ideal gas law assumes no intermolecular forces and that gas particles occupy no volume, while the van der Waals equation accounts for these factors, making it more accurate for real gases at high pressures.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law assumes that gas particles do not interact and occupy no volume, making it a useful approximation under many conditions.
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Ideal Gas Law Formula
van der Waals Equation
The van der Waals equation is an adjustment of the Ideal Gas Law that accounts for the finite size of gas molecules and the attractive forces between them. It is expressed as (P + a(n/V)²)(V - nb) = nRT, where 'a' and 'b' are constants specific to each gas. This equation provides a more accurate description of real gas behavior, especially at high pressures and low temperatures, where deviations from ideality are significant.
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Real vs. Ideal Gas Behavior
Real gases deviate from ideal behavior due to intermolecular forces and the volume occupied by gas molecules. At high pressures and low temperatures, these effects become pronounced, leading to differences in calculated volumes when using the Ideal Gas Law versus the van der Waals equation. Understanding these differences is crucial for accurately predicting gas behavior in various conditions, as real gases do not always conform to the assumptions of ideality.
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Related Practice
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