# Why is estimation of ##\frac{Pv}{RT}=1+BP+CP^2+...## interesting?

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• zenterix
zenterix
TL;DR Summary
There is a derivation in the book Heat and Thermodynamics by Zemansky and Dittman which links the ideal gas law to an empirical power series equation relating ##Pv## and ##P##. The final equation of the derivation is ##\frac{Pv}{RT}=1+BP+CP^2+...## and there is a table with the estimated virial coefficients for nitrogen gas. I would like to know what makes this particular form of the equation interesting?
Consider ##n## moles of a gas at a constant temperature ##T##.

If we vary pressure ##P## and measure the corresponding values of volume ##V##, we can make a plot of ##P\frac{V}{n}=Pv## against ##P##.

This gives us some graph which has some form. Turns out that for a range of pressure starting at 0 the graph is approximately linear. At higher pressures, it becomes more nonlinear.

We can model this relationship using a power series

$$Pv=A(1+BP+CP^2+...)\tag{1}$$

Empirically, we see that for any such plot (ie, for any gas), the vertical intercept is the same. That is, ##A## is the same in the power series.

We can find what this limiting value is from the ideal gas law. For a constant volume gas, we have

$$T=273.16\text{K} \cdot \lim\limits_{P_{TP}\to 0} \left ( \frac{P}{P_{TP}} \right )\tag{2}$$

$$=273.16\text{K} \frac{\lim\limits_{P_{TP}\to 0} Pv}{\lim\limits_{P_{TP}\to 0} P_{TP}v}\tag{3}$$

and so

$$\lim\limits_{P_{TP}\to 0} Pv = \frac{\lim\limits_{P_{TP}\to 0} P_{TP}v}{273.16\text{K}}\cdot T\tag{4}$$

$$=RT$$

where $R$ is the molar gas constant.

We can also write

$$\lim\limits_{P_{TP}\to 0} PV=nRT\tag{5}$$

which is an equation of state for a gas in a hypothetical limit of low pressure.

Now, let's go back to the idea of modeling the entire relationship between ##Pv## and ##P##.

$$Pv=A(1+BP+CP^2+...)=RT(1+BP+CP^2+...)\tag{6}$$

$$\frac{Pv}{RT}=1+BP+CP^2+...\tag{7}$$

My question is about equation (7). Essentially, why is it interesting in this form?

Consider the following table

Here we have estimates for the virial coefficients in (7) for nitrogen gas.

Why is the formulation in (7) interesting?

Here is my attempt to answer this
- If the gas were ideal then we would have ##\frac{Pv}{RT}=1##. That is, ##B=C=D=...=0##.

- But we are dealing with a real gas.

- For a fixed temperature, if the value of ##\frac{Pv}{RT}## is above 1 then I assume this means that the molar volume is larger than expected for an ideal gas. I remember reading something in the past about intermolecular forces. Is this the origin of such a discrepancy at higher pressures?

- Below, I take the values of the table above for three temperatures and plot equation (7).

Thus, it seems that (7) allows us to gauge deviation from ideal gas behavior in a relatively simple way, namely deviation from 1.

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zenterix said:
TL;DR Summary: There is a derivation in the book Heat and Thermodynamics by Zemansky and Dittman which links the ideal gas law to an empirical power series equation relating ##Pv## and ##P##. The final equation of the derivation is ##\frac{Pv}{RT}=1+BP+CP^2+...## and there is a table with the estimated virial coefficients for nitrogen gas. I would like to know what makes this particular form of the equation interesting?

- For a fixed temperature, if the value of PvRT is above 1 then I assume this means that the molar volume is larger than expected for an ideal gas. I remember reading something in the past about intermolecular forces. Is this the origin of such a discrepancy at higher pressures?
Ideal gas has no molecule volume. Real gas molecules have volume which makes upward tendency in the graph. Some real gas molecules have in general attractive forces between which makes downward tendency.

zenterix said:
TL;DR Summary: There is a derivation in the book Heat and Thermodynamics by Zemansky and Dittman which links the ideal gas law to an empirical power series equation relating ##Pv## and ##P##. The final equation of the derivation is ##\frac{Pv}{RT}=1+BP+CP^2+...## and there is a table with the estimated virial coefficients for nitrogen gas. I would like to know what makes this particular form of the equation interesting?

Consider ##n## moles of a gas at a constant temperature ##T##.

If we vary pressure ##P## and measure the corresponding values of volume ##V##, we can make a plot of ##P\frac{V}{n}=Pv## against ##P##.

This gives us some graph which has some form. Turns out that for a range of pressure starting at 0 the graph is approximately linear. At higher pressures, it becomes more nonlinear.

We can model this relationship using a power series

$$Pv=A(1+BP+CP^2+...)\tag{1}$$

Empirically, we see that for any such plot (ie, for any gas), the vertical intercept is the same. That is, ##A## is the same in the power series.

We can find what this limiting value is from the ideal gas law. For a constant volume gas, we have

$$T=273.16\text{K} \cdot \lim\limits_{P_{TP}\to 0} \left ( \frac{P}{P_{TP}} \right )\tag{2}$$

$$=273.16\text{K} \frac{\lim\limits_{P_{TP}\to 0} Pv}{\lim\limits_{P_{TP}\to 0} P_{TP}v}\tag{3}$$

and so

$$\lim\limits_{P_{TP}\to 0} Pv = \frac{\lim\limits_{P_{TP}\to 0} P_{TP}v}{273.16\text{K}}\cdot T\tag{4}$$

$$=RT$$

where $R$ is the molar gas constant.

We can also write

$$\lim\limits_{P_{TP}\to 0} PV=nRT\tag{5}$$

which is an equation of state for a gas in a hypothetical limit of low pressure.

Now, let's go back to the idea of modeling the entire relationship between ##Pv## and ##P##.

$$Pv=A(1+BP+CP^2+...)=RT(1+BP+CP^2+...)\tag{6}$$

$$\frac{Pv}{RT}=1+BP+CP^2+...\tag{7}$$

My question is about equation (7). Essentially, why is it interesting in this form?

Consider the following table

View attachment 335216

Here we have estimates for the virial coefficients in (7) for nitrogen gas.

Why is the formulation in (7) interesting?

Here is my attempt to answer this
- If the gas were ideal then we would have ##\frac{Pv}{RT}=1##. That is, ##B=C=D=...=0##.

- But we are dealing with a real gas.

- For a fixed temperature, if the value of ##\frac{Pv}{RT}## is above 1 then I assume this means that the molar volume is larger than expected for an ideal gas. I remember reading something in the past about intermolecular forces. Is this the origin of such a discrepancy at higher pressures?

- Below, I take the values of the table above for three temperatures and plot equation (7).

View attachment 335217

Thus, it seems that (7) allows us to gauge deviation from ideal gas behavior in a relatively simple way, namely deviation from 1.
Yes. Equation 7 is a relationship for calculating the "compressibility factor" z.

## Why is the estimation of ##\frac{Pv}{RT}=1+BP+CP^2+...## interesting in thermodynamics?

This equation, known as the virial equation of state, provides a more accurate representation of the behavior of real gases compared to the ideal gas law. It accounts for intermolecular forces and the finite size of molecules, which are significant at high pressures and low temperatures.

## What do the coefficients B, C, etc., represent in the virial equation?

The coefficients B, C, etc., are known as virial coefficients. They are temperature-dependent parameters that quantify the interactions between molecules in a gas. The second virial coefficient (B) accounts for pairwise interactions, the third (C) for three-body interactions, and so on.

## How are virial coefficients determined experimentally?

Virial coefficients are typically determined through experimental measurements of pressure, volume, and temperature for a real gas. Data from these measurements are then fitted to the virial equation to extract the coefficients. Techniques such as isothermal compressibility and sound velocity measurements can be used.

## Why is the virial equation preferred over the ideal gas law for real gases?

The ideal gas law assumes no intermolecular forces and infinite molecular size, which is not true for real gases. The virial equation corrects for these assumptions by incorporating terms that account for molecular interactions and volume, providing a more accurate description of gas behavior under various conditions.

## Can the virial equation be used for all gases and conditions?

While the virial equation provides a better approximation than the ideal gas law, it has limitations. It is most accurate for gases at moderate pressures and temperatures. At very high pressures or very low temperatures, more complex equations of state, such as the van der Waals or Redlich-Kwong equations, might be required to accurately describe gas behavior.

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