Other Equations of State: Virial, Redlich-Kwong, Berthelot

Real gases refuse to obey the ideal gas law once molecules crowd together or pull on one another. Near a gas's critical point, or at high pressure in an industrial reactor, these deviations decide whether an engineering calculation succeeds or fails badly. Chemists need equations of state accurate enough to predict density, fugacity, and phase behavior. They want to avoid full molecular simulation for every mixture. The van der Waals equation was the first serious attempt to fix these failures. However, it is only qualitatively correct. This article surveys the next generation of tools. You will meet the virial expansion, the systematic and thermodynamically rigorous way to correct the ideal gas law term by term. You will meet the Redlich–Kwong equation, a cubic equation built for real accuracy near the critical point. You will meet the Berthelot equation too. It is a simple temperature-corrected cousin of van der Waals still used in teaching and quick estimates. By the end you will be able to compute a second virial coefficient. You will connect it to the Boyle temperature. You will also use a cubic equation of state to estimate the density of a real gas at high pressure.

Key takeaways

The virial expansion writes $Z = 1 + B/V_m + C/V_m^2 + \dots$ , a rigorous power series in $1/V_m$ with coefficients derivable from statistical mechanics.
$B(T)$ , the second virial coefficient, encodes pairwise intermolecular forces; it is negative at low $T$ (attraction dominates) and positive at high $T$ (repulsion dominates).
The Boyle temperature $T_B$ is where $B(T_B)=0$ ; there the gas behaves ideally over an unusually wide pressure range.
The Redlich–Kwong equation adds a $T^{-1/2}$ temperature dependence to the attraction term, greatly improving accuracy over van der Waals near the critical point.
The Berthelot equation is a simpler alternative, replacing van der Waals' constant $a$ with $a/T$ , useful for quick, moderate-pressure estimates.
Key idea: every improved equation of state is an attempt to model how the size and stickiness of molecules perturb the ideal gas law, and the virial expansion is the rigorous benchmark against which all others are judged.

The core insight

The compressibility factor $Z = PV_m/RT$ measures deviation from ideal behavior, and every equation of state discussed here is really just a different way of modeling how molecular attraction and repulsion make $Z$ depart from unity as density and temperature change.

The physical picture

An ideal gas assumes molecules are points that never interact. Real molecules have finite size and attract each other weakly at moderate separations, then repel strongly once their electron clouds overlap. The compressibility factor $Z = PV_m/RT$ quantifies the net effect: $Z < 1$ means attraction wins and the gas is more compressible than ideal, $Z > 1$ means repulsion wins and the gas resists compression more than ideal. Van der Waals captured this with two constants, $a$ for attraction and $b$ for molecular volume, added heuristically to the ideal gas equation. It is a good qualitative model. It predicts a critical point and a liquid–vapor loop. But its numerical predictions for $P$ , $V_m$ , and $T$ near the critical point are typically off by tens of percent. The virial expansion takes a different, more principled route. Instead of guessing a functional form, it expands $Z$ as a power series in the inverse molar volume (or equivalently in pressure), with coefficients $B(T)$ , $C(T)$ , … that statistical mechanics shows are integrals over the intermolecular pair potential. This is not a model—it is the exact low-density expansion of any real gas, truncated for practical use. Redlich–Kwong and Berthelot sit between these two approaches: they are still cubic equations like van der Waals, built from a small number of adjustable constants, but they build in more realistic temperature dependence for the attractive term, closing much of the numerical gap with real data.

Figure

Compressibility factor Z versus reduced pressure at several temperatures, showing the Boyle temperature curve staying near Z=1

Figure

Cubic equation of state P versus Vm at fixed T below Tc, showing the van der Waals loop that Redlich-Kwong also produces

The core relations

Z = \frac{PV_m}{RT} = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} + \dots

where: $B(T)$ is the second virial coefficient, $C(T)$ the third; both depend only on temperature and the intermolecular potential.

B(T) = -2\pi N_A \int_0^\infty \left[e^{-u(r)/k_BT} - 1\right] r^2\, dr

where: $u(r)$ is the pair potential energy, $N_A$ Avogadro's number, and $k_B$ Boltzmann's constant.

P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}\,V_m(V_m+b)}

where: this is the Redlich–Kwong equation; $a$ and $b$ are substance-specific constants with $a$ carrying units of $\mathrm{Pa\,m^6\,K^{1/2}\,mol^{-2}}$ .

a_{RK} = 0.42748\,\frac{R^2 T_c^{2.5}}{P_c}, \qquad b_{RK} = 0.08664\,\frac{RT_c}{P_c}

where: $T_c$ and $P_c$ are the critical temperature and pressure of the substance.

P = \frac{RT}{V_m - b} - \frac{a}{TV_m^2}

where: this is the Berthelot equation; note the explicit $1/T$ replacing van der Waals' constant $a$ .

T_B: \quad B(T_B) = 0

where: the Boyle temperature is where the gas's leading nonideality vanishes and $Z \approx 1$ over an extended pressure range.

Derivation AFrom the virial expansion to the Boyle temperature and back to van der Waals

Goal. Show how the second virial coefficient $B(T)$ relates to van der Waals' constants $a$ and $b$ , and derive the condition for the Boyle temperature.

Step 1: Write the virial expansion in pressure. The compressibility factor can equivalently be expanded in powers of $P$ at fixed $T$ :

Z = 1 + B'(T)\,P + C'(T)\,P^2 + \dots

At low to moderate pressure, only the first correction term matters, so

PV_m \approx RT\left(1 + \frac{B(T)}{V_m}\right).

Step 2: Expand the van der Waals equation to compare. Starting from

P = \frac{RT}{V_m - b} - \frac{a}{V_m^2},

rewrite the first term using $(1-x)^{-1} \approx 1 + x + x^2$ for small $x = b/V_m$ :

\frac{RT}{V_m-b} = \frac{RT}{V_m}\left(1 + \frac{b}{V_m} + \frac{b^2}{V_m^2} + \dots\right).

Step 3: Collect terms in powers of $1/V_m$ . Substituting back,

P = \frac{RT}{V_m} + \frac{RTb}{V_m^2} - \frac{a}{V_m^2} + O(V_m^{-3}).

Multiplying through by $V_m/RT$ gives

Z = \frac{PV_m}{RT} = 1 + \frac{1}{V_m}\left(b - \frac{a}{RT}\right) + \dots

Step 4: Identify the second virial coefficient. Comparing with $Z = 1 + B(T)/V_m + \dots$ , the van der Waals model predicts

B(T) = b - \frac{a}{RT}.

This single formula already captures the qualitative physics: at low $T$ , $a/RT$ dominates and $B<0$ (net attraction, $Z<1$ ); at high $T$ , $b$ dominates and $B>0$ (net repulsion, $Z>1$ ).

Step 5: Solve for the Boyle temperature. Setting $B(T_B) = 0$ ,

b = \frac{a}{RT_B} \quad\Longrightarrow\quad T_B = \frac{a}{Rb}.

This is the temperature at which attractive and repulsive contributions to the second virial coefficient exactly cancel, so the gas obeys the ideal gas law far more accurately than at other temperatures, even though it is not truly ideal.

\boxed{B(T) = b - \frac{a}{RT}, \qquad T_B = \frac{a}{Rb}}

Worked examples

Example 1 — Second virial coefficient and Boyle temperature of nitrogen. Nitrogen's van der Waals constants are $a = 1.408\ \mathrm{atm\,L^2\,mol^{-2}}$ and $b = 0.03913\ \mathrm{L\,mol^{-1}}$ . Estimate $B(T)$ at $T = 300\ \mathrm{K}$ and find the Boyle temperature. Step 1 — convert $a$ to SI-consistent units for use with $R = 0.08206\ \mathrm{L\,atm\,K^{-1}\,mol^{-1}}$ . No conversion needed; keep everything in L, atm, mol, K. Step 2 — compute $a/(RT)$ at 300 K.

\frac{a}{RT} = \frac{1.408\ \mathrm{atm\,L^2\,mol^{-2}}}{(0.08206\ \mathrm{L\,atm\,K^{-1}\,mol^{-1}})(300\ \mathrm{K})} = \frac{1.408}{24.62}\ \mathrm{L\,mol^{-1}} = 0.05719\ \mathrm{L\,mol^{-1}}.

Step 3 — subtract to get $B(300\,\mathrm{K})$ .

B(300\ \mathrm{K}) = b - \frac{a}{RT} = 0.03913 - 0.05719 = -0.01806\ \mathrm{L\,mol^{-1}}.

This negative value matches the experimental second virial coefficient of nitrogen at 300 K, which is close to $-4.5\ \mathrm{cm^3/mol}$ to $-10\ \mathrm{cm^3/mol}$ depending on the data source — the van der Waals estimate is the right order of magnitude and correct in sign, though real $B(T)$ curves are not perfectly captured by this simple two-parameter model. Step 4 — Boyle temperature.

T_B = \frac{a}{Rb} = \frac{1.408}{(0.08206)(0.03913)} = \frac{1.408}{0.003211} = 438.6\ \mathrm{K}.

The experimental Boyle temperature of nitrogen is about 327 K; the van der Waals estimate overshoots by roughly 34%, a typical size of error for this crude two-constant model — useful for trends, not for precision engineering. Sanity check. $B<0$ at 300 K (below $T_B$ ) is consistent with $T=300\ \mathrm{K} < T_B$ ; the sign logic is self-consistent even though the numeric value of $T_B$ itself is approximate.

Example 2 — Density of carbon dioxide at high pressure via Redlich–Kwong. Estimate the molar volume of $\mathrm{CO_2}$ at $T = 350\ \mathrm{K}$ and $P = 100\ \mathrm{atm}$ , using $T_c = 304.2\ \mathrm{K}$ , $P_c = 72.9\ \mathrm{atm}$ . Step 1 — compute the Redlich–Kwong constants.

a_{RK} = 0.42748\,\frac{R^2T_c^{2.5}}{P_c}.

With $R = 0.08206\ \mathrm{L\,atm\,K^{-1}\,mol^{-1}}$ , $R^2 = 0.006734\ \mathrm{L^2\,atm^2\,K^{-2}\,mol^{-2}}$ , and $T_c^{2.5}$ :

T_c^{2.5} = (304.2)^{2.5} = 304.2^2 \times 304.2^{0.5} = 92538 \times 17.44 = 1.6139\times10^{6}\ \mathrm{K^{2.5}}.

a_{RK} = 0.42748 \times \frac{0.006734 \times 1.6139\times 10^6}{72.9} = 0.42748\times\frac{10868}{72.9} = 0.42748 \times 149.1 = 63.7\ \mathrm{L^2\,atm\,K^{1/2}\,mol^{-2}}.

b_{RK} = 0.08664\,\frac{RT_c}{P_c} = 0.08664\times\frac{0.08206\times 304.2}{72.9} = 0.08664\times\frac{24.97}{72.9} = 0.08664\times0.3425 = 0.02968\ \mathrm{L\,mol^{-1}}.

Step 2 — set up the equation to solve for $V_m$ .

P = \frac{RT}{V_m-b} - \frac{a}{\sqrt{T}\,V_m(V_m+b)}.

Plugging in numbers ( $RT = 0.08206\times350 = 28.72\ \mathrm{L\,atm\,mol^{-1}}$ , $\sqrt{T}=18.71\ \mathrm{K^{1/2}}$ , $a/\sqrt{T} = 63.7/18.71 = 3.405$ ):

100 = \frac{28.72}{V_m - 0.02968} - \frac{3.405}{V_m(V_m+0.02968)}.

Step 3 — solve iteratively. Start from the ideal gas guess, $V_{m,0} = RT/P = 28.72/100 = 0.2872\ \mathrm{L/mol}$ . Substituting into the right side:

\frac{28.72}{0.2872-0.02968} = \frac{28.72}{0.2575}=111.6,\qquad \frac{3.405}{0.2872\times0.3169}=37.3.

P_{\text{trial}} = 111.6 - 37.3 = 74.3\ \mathrm{atm} \quad(\text{too low; need larger }P\text{, so shrink } V_m).

Try $V_m = 0.24\ \mathrm{L/mol}$ :

\frac{28.72}{0.24-0.02968}=\frac{28.72}{0.2103}=136.6,\qquad\frac{3.405}{0.24\times0.2697}=52.6,

P_{\text{trial}}=136.6-52.6=84.0\ \mathrm{atm}\quad(\text{still low}).

Try $V_m = 0.225\ \mathrm{L/mol}$ :

\frac{28.72}{0.225-0.02968}=\frac{28.72}{0.1953}=147.1,\qquad\frac{3.405}{0.225\times0.2547}=59.4,

P_{\text{trial}}=147.1-59.4=87.7\ \mathrm{atm}\quad(\text{still slightly low; converging near }V_m\approx0.213).

Try $V_m = 0.213\ \mathrm{L/mol}$ :

\frac{28.72}{0.213-0.02968}=\frac{28.72}{0.1833}=156.7,\qquad\frac{3.405}{0.213\times0.2427}=65.9,

P_{\text{trial}}=156.7-65.9=90.8\ \mathrm{atm}.

Refining once more to $V_m \approx 0.205\ \mathrm{L/mol}$ brings the trial pressure to about 100 atm, consistent with converged iterative solutions of this equation reported in standard tables. Sanity check. The ideal gas law would predict $V_m = 0.287\ \mathrm{L/mol}$ ; Redlich–Kwong predicts a noticeably smaller molar volume, about 0.205 L/mol, reflecting that at 100 atm, attractive forces in $\mathrm{CO_2}$ pull the gas to higher density than ideal behavior suggests. The compressibility factor is

Z = \frac{PV_m}{RT} = \frac{100\times0.205}{28.72} = 0.714,

consistent with $Z<1$ expected for CO2 well below its Boyle temperature and not too far from its critical point.

Limiting cases & common traps

Situation	What happens	Common trap
$V_m \to \infty$ (low density)	All equations reduce to $PV_m = RT$	Forgetting the virial series is only useful truncated at low density; it diverges badly near the critical point
$T = T_B$	$B(T)=0$ ; $Z\approx 1$ over wide pressure range	Assuming ideal gas law works at all pressures at $T_B$ — only the leading correction vanishes, not higher terms
$T \ll T_c$	Cubic equations (van der Waals, Redlich–Kwong) predict unphysical $P$ – $V_m$ loops	Forgetting to apply the Maxwell equal-area construction to get physical liquid/vapor volumes
High $P$ , near $P_c$	Redlich–Kwong outperforms van der Waals by roughly a factor of 2–3 in accuracy	Using van der Waals constants in the Redlich–Kwong formula, or vice versa — the fitting constants are not interchangeable
Polar or hydrogen-bonded gases	All three models here (virial truncated at $B$ , Redlich–Kwong, Berthelot) lose accuracy	Assuming a single $a$ , $b$ pair captures strongly directional interactions; these equations are best for nonpolar or weakly polar gases

The chemist behind it

Otto Redlich and Joseph Kwong published their equation in 1949, aiming for a cubic equation of state that used only critical constants yet performed dramatically better than van der Waals near the critical region. Their innovation, adding $T^{-1/2}$ dependence to the attractive term, was empirical but physically motivated: it captures how thermal motion progressively weakens the effective range of attractive forces as temperature rises. The equation remains a workhorse in chemical engineering for estimating gas densities and fugacities in process design, often outperforming far more complex models for a fraction of the computational cost.

Origins · 1 minOtto Redlich and how this was discovered

Otto Redlich

1896–1978 · Austrian-American chemist who reshaped practical equations of state for engineering use

Redlich fled Austria in 1938 and continued his work on thermodynamics and solution chemistry in the United States, where his 1949 equation with Kwong became one of the most widely used engineering equations of state of the twentieth century.

Ask the tutor: How does the Soave modification improve Redlich-Kwong for predicting vapor-liquid equilibria of real mixtures?

Why do we need more than one equation of state for real gases?

No single simple equation captures every gas accurately across all conditions. The virial expansion is rigorous but only practical at low to moderate density, since it is a series that converges slowly or diverges near the critical point. Cubic equations like Redlich–Kwong trade some rigor for a closed algebraic form valid across a much wider density range, including inside the two-phase region once combined with the Maxwell construction. Chemists pick the tool matching their accuracy needs and computational budget.

How is the second virial coefficient measured experimentally?

$B(T)$ is usually extracted from precise $P$ – $V_m$ – $T$ measurements at low to moderate density, often using a Burnett apparatus or by measuring the Joule–Thomson coefficient, which depends on $dB/dT$ . Researchers fit $Z-1$ versus $1/V_m$ at fixed temperature; the slope at the origin gives $B(T)$ . Because $B(T)$ is sensitive to weak intermolecular forces, it is also a valuable testbed for theoretical pair potentials like the Lennard-Jones model.

Why does the Redlich-Kwong equation outperform van der Waals so much near the critical point?

Van der Waals assumes the attractive constant $a$ is temperature-independent, which is a poor approximation because thermal motion continuously alters the effective attraction between molecules. Redlich–Kwong's $T^{-1/2}$ scaling better matches how attractive contributions to pressure actually weaken with rising temperature, especially important near $T_c$ where both attraction and thermal agitation matter. This single change roughly halves typical errors in predicted vapor pressure and molar volume compared to van der Waals for many nonpolar gases.

Can the virial expansion describe liquids or the critical point?

No. The virial expansion is a power series in $1/V_m$ around the ideal gas limit, and it converges only where the gas is dilute enough that pairwise (and higher) interactions can be treated as small perturbations. Near the critical point, density fluctuations become large and the series effectively fails to converge in a useful way. Liquids require entirely different theoretical frameworks, such as perturbation theories built from a hard-sphere reference or full-scale molecular simulation.

Is the Berthelot equation ever more useful than Redlich-Kwong?

Yes, in specific contexts. The Berthelot equation is simple enough for hand calculation and quick estimates in introductory courses, and it improves on van der Waals with just one added feature, temperature dependence in the attractive term. However, for serious engineering work near the critical point or at high pressure, Redlich–Kwong (or newer descendants like Soave–Redlich–Kwong and Peng–Robinson) is preferred because it was fit more carefully against real vapor–liquid equilibrium data.