Question

The virial form of van der Waal's gas equation is \(P V=R T\left(1+\frac{B}{V}+\frac{C}{V^{2}}+\ldots\right)\) \(=R T\left(1+B^{\prime} P+C^{\prime} P^{2}+\ldots\right) .\) The sec- ond virial coefficient of argon gas at \(262.5 \mathrm{~K}\) is \(-11 \mathrm{~mol}^{-1}\). What is the density of argon gas at \(262.5 \mathrm{~K}\) and \(1 \mathrm{~atm}\) ? Neglect all the terms after second term in the virial forms, under these condition. \((R=0.08 \mathrm{~L}-\mathrm{atm} / \mathrm{K}-\mathrm{mol}, \mathrm{Ar}=40)\) (a) \(2.0 \mathrm{~g} / 1\) (b) \(1.905 \mathrm{~g} / \mathrm{l}\) (c) \(1.818 \mathrm{~g} / 1\) (d) \(1.964 \mathrm{~g} / \mathrm{l}\)

Short answer

The density of argon gas at 262.5 K and 1 atm, after calculating the volume with the simplified virial equation and correcting for the second virial coefficient, is approximately 1.964 g/L (option d).

Step-by-step solution

Understand the Provided Information

We are given the virial form of van der Waal's gas equation, the second virial coefficient for argon gas at a specific temperature, the value of the gas constant R, and the molar mass of argon. We are also instructed to use only the first two terms of the virial expansion.

Simplify the Van der Waals Virial Equation

Neglecting higher-order terms, the virial equation simplifies to: \( P V = RT(1 + B'P) \), where \( B' \) is related to the second virial coefficient.

Relate the Second Virial Coefficient to Pressure

From the given second virial coefficient \( B = -11 L/mol \) and knowing that \( P V = nRT \), we can derive the expression for \( B' \) as \( B' = \frac{B}{RT} \).

Calculate the Value of \( B' \)

Substitute the values for \( B \), \( R \), and \( T \) into the expression for \( B' \): \( B' = \frac{-11}{0.082 * 262.5} \).

Insert \( B' \) into the Simplified Virial Equation

Use the calculated value of \( B' \) and the given pressure of 1 atm to find the volume: \( P V = RT(1 + B'P) \) becomes \( V = \frac{RT}{P}(1 + B'P) \).

Calculate the Density of Argon Gas

The density (\( \rho \)) is the mass per unit volume, which can be calculated using the molar mass of argon, \( M_{Ar} = 40 g/mol \), and the molar volume \( V \) found in the previous step: \( \rho = \frac{M_{Ar}}{V} \).

Find the Mass of 1 Mole of Argon Gas in the Calculated Volume

Using the molar mass of argon, we know that 1 mole weighs 40 g. We then use the volume per mole calculated to find the density.

Key concepts

Van der Waals Gas Equation

The van der Waals gas equation is a sophisticated way to describe the behavior of real gases, taking into account the finite size of gas molecules and the attractive forces between them. Unlike the ideal gas law, which assumes point-like particles and no intermolecular attractions, the van der Waals equation introduces corrections that account for the volume occupied by gas molecules (\textbf{volume correction}) and the pressure decrease due to attractions (\textbf{pressure correction}). Thus, it allows for more accurate predictions of gas behavior under various conditions.

The general form of the van der Waals equation is \( (P + a(n/V)^2)(V - nb) = nRT \), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, \(T\) is the temperature, \(a\) is a measure of the attraction between particles, and \(b\) is the volume occupied by one mole of the particles. For practical calculations at low pressures and high temperatures where gases behave more ideally, the virial form of the equation, which includes a series expansion to correct the behavior, is often used.

Second Virial Coefficient

The second virial coefficient \(B\) provides a quantitative measure of the deviations from ideal gas behaviors in terms of molecular interactions. It appears in the virial equation of state, which is an infinite power series expansion to predict the behavior of real gases with higher accuracy. The second virial coefficient is temperature-dependent, reflects attractive or repulsive forces, and is typically the first correction term after the ideal gas term.

The sign of the second virial coefficient can tell us about the nature of intermolecular forces: a negative \(B\) indicates dominating attractive forces, while a positive \(B\) implies that repulsive forces are significant. In the virial expansion form used in the exercise, \(B\) affects the pressure and volume relationship in the simplified equation to retain the first-order non-ideal effects.

Equation of State

An equation of state (EoS) is a mathematical model that describes the state of matter under a given set of physical conditions. It provides a relationship between state variables like pressure, volume, and temperature. For gases, the most simplified EoS is the Ideal Gas Law, \(PV = nRT\), which assumes no interaction between gas molecules and that they occupy no volume. However, real gases deviate from this ideal behavior, leading to the development of more complex equations like the van der Waals equation and the virial equation.

These real gas equations incorporate additional factors such as intermolecular forces and the volume occupied by gas molecules themselves to refine the description of the gas state. The virial equation of state generalizes these concepts by using series expansions with coefficients (like the second virial coefficient) that adjust the calculations to better match real gas behavior.

Gas Density Calculations

Calculating the density of a gas involves finding the mass of the gas per unit volume. Under the framework of the Ideal Gas Law, the density (\textbf{\text{\rho}}) can simply be obtained from \textbf{\text{\rho}} = \(\frac{m}{V}\) where \(m\) is the mass and \(V\) is the volume. In real-gas scenarios, the virial equation of state is employed. To bring this into a practical context as per the exercise, once we have the virial expansion simplified to consider only the second virial coefficient, the real volume of the gas can be calculated. Using this volume and the molar mass of the gas (molar weight for argon in this case), we determine the gas density.

Through this method, the calculation accounts for non-ideal gas behavior at a given temperature and pressure, giving a more accurate estimation of density compared to assuming ideal gas behavior. This kind of precise calculation is crucial for many practical applications in engineering and the physical sciences where precise measurements of gas density are required.