Question

Which of the following gases has the maximum value of \(^{4} \mathrm{a}\) " in Van der Waals equation? (1) \(\mathrm{H}_{2}\) (2) \(\mathrm{He}\) (3) \(\mathrm{O}_{2}\) (4) \(\mathrm{NH}_{3}\)

Short answer

NH\textsubscript{3} has the maximum value of \(a\).

Step-by-step solution

Understand the Van der Waals Equation

The Van der Waals equation is given by \[ \bigg( P + \frac{a}{V_m^2} \bigg)(V_m - b) = RT \]where \(a\) represents the measure of the attraction between molecules, and \(b\) is the volume excluded by a mole of the gas particles.

Identify the Variables related to Strength of Attraction

Among the constants, a higher value of the Van der Waals constant \(a\) indicates stronger intermolecular forces of attraction.

Compare the given gases

Now, let's compare the intermolecular attraction (\(a\) values) for the gases listed: \(\mathrm{H_{2}}: a = 0.244 \, L^2 bar/mol^2\) \(\mathrm{He}: a = 0.0341 \, L^2 bar/mol^2\) \(\mathrm{O_{2}}: a = 1.360 \, L^2 bar/mol^2\) \(\mathrm{NH_{3}}: a = 4.170 \, L^2 bar/mol^2\)

Determine the Maximum Value

Among the given gases, \( \mathrm{NH_{3}} \) has the highest value of the Van der Waals constant \(a = 4.170 \, L^2 bar/mol^2\).

Key concepts

Van der Waals Equation and Intermolecular Forces

The Van der Waals equation is a modified version of the Ideal Gas Law to account for intermolecular forces and the finite size of gas molecules. It is given by:equationdisplaying:\(\big( P + \frac{a}{V_m^2} \big)(V_m - b) = RT\)In this equation, \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the universal gas constant, and \(T\) is temperature. The constants \(a\) and \(b\) correct for intermolecular attractions and the volume occupied by gas molecules, respectively. The constant \(a\), in particular, signifies the strength of intermolecular attractions. A higher \(a\) value indicates stronger attraction between molecules. For example, among \(H_2\), \(He\), \(O_2\), and \(NH_3\), \(NH_3\) exhibits the strongest intermolecular forces because it has the highest \(a\) value.

Ideal Gas Law and Its Limitations

The Ideal Gas Law is expressed as:\(PV = nRT\)where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature. This equation assumes that gas molecules have no volume and do not interact with each other through intermolecular forces. Although simple, it falls short when dealing with real gases, especially under high pressure or low temperature. That’s where the Van der Waals equation comes in, precisely introducing parameters \(a\) and \(b\) to correct for these interactions and the non-zero size of molecules, thus providing more accurate predictions.

Understanding Gas Constants

In gas laws, various constants are used for precise measurements and predictions. The universal gas constant \(R\) is a key part of these laws. It is a physical constant expressed as \(8.314\ J/(mol·K)\) and ties together multiple gas law equations, like the Ideal Gas Law and the Van der Waals Equation.Additionally, in the Van der Waals equation, two unique constants \(a\) and \(b\) are tailored for each gas:

• \(a\): Measures intermolecular attraction. Higher values indicate stronger forces, as seen in gases like \(NH_3\).
• \(b\): Accounts for the volume occupied by the gas particles themselves.

Understanding these constants helps in predicting the behavior of real gases under various conditions. Comparing these constants across different gases reveals their unique properties, such as why \(NH_3\) has more intermolecular forces compared to \(H_2\) or \(He\).