Problem 17 - An Introduction to Thermal Physics

Question

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion, $$P V=n R T\left(1+\frac{B(T)}{(V / n)}+\frac{C(T)}{(V / n)^{2}}+\cdots\right)$$ where the functions \(B(T), C(T),\) and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it's sufficient to omit the third term and concentrate on the second, whose coefficient \(B(T)\) is called the second virial coefficient (the first coefficient being 1 ). Here are some measured values of the second virial coefficient for nitrogen \(\left(\mathrm{N}_{2}\right):\) $$\begin{array}{cc} T(\mathrm{K}) & B\left(\mathrm{cm}^{3} / \mathrm{mol}\right) \\ \hline 100 & -160 \\ 200 & -35 \\ 300 & -4.2 \\ 400 & 9.0 \\ 500 & 16.9 \\ 600 & 21.3 \end{array}$$ (a) For each temperature in the table, compute the second term in the virial equation, \(B(T) /(V / n),\) for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions. (b) Think about the forces between molecules, and explain why we might expect \(B(T)\) to be negative at low temperatures but positive at high temperatures. (c) Any proposed relation between \(P, V,\) and \(T,\) like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation, $$\left(P+\frac{a n^{2}}{V^{2}}\right)(V-n b)=n R T$$ where \(a\) and \(b\) are constants that depend on the type of gas. Calculate the second and third virial coefficients \((B \text { and } C)\) for a gas obeying the van der Waals equation, in terms of \(a\) and \(b\). (Hint: The binomial expansion says that \((1+x)^{p} \approx 1+p x+\frac{1}{2} p(p-1) x^{2},\) provided that \(|p x| \ll 1 .\) Apply this approximation to the quantity \([1-(n b / V)]^{-1}\).) (d) Plot a graph of the van der Waals prediction for \(B(T),\) choosing \(a\) and \(b\) so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Step-by-Step Solution

Answer

The ideal gas law better approximates real gas behavior at higher temperatures for nitrogen. Attractive forces dominate at low temperatures causing negative virial coefficients.

Step 1: Understanding the Virial Expansion

The virial expansion provides a way to account for deviations from the ideal gas law. For low densities, only the second virial coefficient \(B(T)\) is considered. The equation is \(P V=n R T\left(1+\frac{B(T)}{(V / n)}\right)\).

Step 2: Calculating Second Term for Given Temperatures

For nitrogen at atmospheric pressure and given temperatures, the second term is \(B(T)/ (V / n)\). At \(P = 1\) atm and using the ideal gas law \(PV = nRT\), we compute: - 100 K: \( -160/ 22.4 = -7.14\) - 200 K: \( -35/ 11.2 = -3.13\) - 300 K: \( -4.2/ 7.47 = -0.56\) - 400 K: \( 9/ 5.6 = 1.61\) - 500 K: \( 16.9/ 4.48 = 3.77\) - 600 K: \( 21.3/ 3.73 = 5.71\).

Step 3: Discussing Ideal Gas Law Validity

At lower temperatures, the deviation from ideal behavior (the second term) is more significant, suggesting the ideal gas law is less valid. At higher temperatures, this term is smaller, indicating the ideal gas law is a better approximation.

Step 4: Analyzing Molecular Forces and B(T) Sign

Negative \(B(T)\) at low temperatures suggests attractive forces dominate. Positive \(B(T)\) at high temperatures indicates repulsive forces are more significant. Hence, \(B(T)\) changes sign due to the balance of intermolecular forces.

Step 5: Virial Coefficients from the van der Waals Equation

Using the van der Waals equation, substitute small \(x = nb/V\) in \((1-x)^{-1} \approx 1+x+x^2\) to expand and find: - Second coefficient \(B(T) = b - a/RT\) - Third coefficient \(C(T) = b^2\).

Step 6: Plotting van der Waals Prediction for B(T)

Choose appropriate values for \(a\) and \(b\) to match given \(B(T)\) data for nitrogen. The plot shows how closely the van der Waals equation predicts \(B(T)\) compared to measured data, evaluating its accuracy over the temperature range.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in understanding how gases behave under various conditions. It is expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.
The Ideal Gas Law assumes that gas molecules do not interact with each other and occupy no volume. This simplification makes it incredibly useful for predicting the behavior of gases in different environments, provided the gas conditions are not too extreme.
However, when gas particles are close together, such as at high pressures or low temperatures, the assumptions of the Ideal Gas Law fall short. This is because the particles do interact, and their own volume becomes significant. Deviations from this ideal behavior are the reason behind the development of more sophisticated equations like the virial expansion.

Second Virial Coefficient

In the virial expansion for real gases, the Second Virial Coefficient, denoted as \(B(T)\), provides a correction to the ideal gas behavior. It accounts for molecular interactions that the Ideal Gas Law ignores.
At low gas densities, the virial expansion is dominated by the terms involving \(B(T)\). The coefficient is temperature-dependent and reflects how these interactions change with temperature. At low temperatures, \(B(T)\) is negative, indicating that attractive forces between molecules are dominant. Conversely, at higher temperatures, \(B(T)\) becomes positive, suggesting that repulsive forces take precedence.
The importance of the Second Virial Coefficient lies in its ability to correct the pressure-volume relationship for real gases, particularly when intermolecular forces cannot be ignored. This coefficient is essential for more accurate predictions of real gas behavior, especially at varying temperatures.

van der Waals Equation

The van der Waals Equation is an improved model over the Ideal Gas Law, designed to accommodate the physical properties of real gases. It is expressed as \((P + \frac{an^2}{V^2})(V-nb) = nRT\), where \(a\) and \(b\) are constants unique to each gas.
The parameter \(a\) adjusts for the attractive forces between gas molecules, while \(b\) corrects for the volume occupied by the molecules themselves. Together, these constants help better predict the behavior of gases, especially under conditions where the Ideal Gas Law fails.
One of the notable applications of the van der Waals Equation is its usage in calculating virial coefficients. Using mathematical expansions, one can derive expressions for these coefficients, such as \(B(T)\) and \(C(T)\), allowing a deeper understanding of molecular interactions in gases.

Equation of State

In chemistry and physics, an equation of state is any mathematical model that describes the state of matter under a set of physical conditions. Both the Ideal Gas Law and the van der Waals Equation are examples of equations of state.
Each equation of state provides insights into the relationships between pressure, volume, and temperature of gases. They are critical for calculations in thermodynamics and fluid dynamics as they help predict how gases will respond to changes in external conditions.
The choice of using the Ideal Gas Law, the virial expansion, or the van der Waals Equation depends on the specific situation, such as the presence of intermolecular forces or the density of the gas. These equations play a fundamental role in advancing our understanding of material properties and are pivotal in various scientific and industrial applications.