What does "limit of zero pressure" mean for ideal gas temperature?

[zenterix]

TL;DR Summary

I'm reading the book "Physical Chemistry" by Silbey, Alberty, and Bawendy.

In the first chapter is a section on the zeroth law of thermodynamics and how this leads to the definition of a temperature scale.

I am having trouble understanding certain steps in the reasoning.

Here is the reasoning.

We have three systems (A, B, and C) each consisting of a certain mass of a different fluid (ie, a gas or a compressible liquid).

For the systems we are considering, we know from experiment that pressure and volume are independent thermodynamic variables and that the intensive state of each system is individually is completely described by just these two variables. That is, if equilibrium is reached at a certain pressure and volume, all the macroscopic properties have certain characteristic values.

Next, we think about what happens when we impose an additional constraint on a system, say A, namely that it is in thermal equilibrium with system C that is in a specified state. What happens is that we no longer have two independent variables, but only one: pressure or volume.

For a given value of pressure $P_A$ there is only one volume $V_A$ at which there will be thermal equilibrium with C. If we plot $P_A$ against $V_A$ the resulting curve is called an isotherm.

When we change the state of C we get a different isotherm.

We label each isotherm with something we denote $\Phi$ and that we call temperature.

Here is the point where I start to have doubts.

The book says

Figure 1.4 illustrates Boyle’s law, which states that $PV=$ constant for a specified amount of gas at a specified temperature. Experimentally, this is strictly true only in the limit of zero pressure.

I don't understand this notion about zero pressure.
If pressure is zero, then the lefthand side is zero, so the righthand side must be zero as well, right?

The book goes on

Charles and Gay-Lussac found that the volume of a gas varies linearly with the temperature at specified pressure when the temperature is measured with a mercury in glass thermometer, for example. Since it would be preferable to have a temperature scale that is independent of the properties of particular materials like mercury and glass, it is better to say that the ratio of the $P_2V_2$ product at temperature $\Phi$ to $P_1V_1$ at temperature $\Phi_1$ depends only on the two temperatures:

$$\frac{P_2V_2}{P_1V_1}=\phi(\Phi_1,\Phi_2)\tag{1}$$

where $\phi$ is an unspecified function. The simplest thing to do is to take the ratio of the $PV$ products to be equal to the ratio of the temperatures, thus defining a temperature scale:

$$\frac{P_2V_2}{P_1V_1}=\frac{T_2}{T_1}\tag{2}$$

or

$$\frac{P_2V_2}{T_2}=\frac{P_1V_1}{T_1}\tag{2}$$

Here we have introduced a new symbol T for the temperature because we have made a specific assumption about the function $\phi$. Equations 1.1 and 1.2 are exact only in the limit of zero pressure, and so T is referred to as the ideal gas temperature.

So again, my question is about this notion of zero pressure.

How can we even have zero pressure in these equations?

 

[zenterix]

One aspect I noticed is that the keyword is limit.

Boyle's law says that $PV=$ constant for a specified amount of gas at a specified temperature when we consider the limit of zero pressure. P and V are not independent in this limit: V must approach infinity.

This doesn't get me very much additional understanding however.

 

[Borek]

Do you know how limits work in math? It is exactly the same line of thinking.

 

[zenterix]

Do you know how limits work in math? It is exactly the same line of thinking.

I do know how limits work. In the limit where pressure approaches zero, what happens to volume?

 

[hutchphd]

The statement refers to the behavior of the temperature scale. The limit assumes the volume to be fixed when the limit is considered (this is a thought exercise). It is a definition.

 

[zenterix]

Here is another derivation I found in another book ("Physical Chemistry" by Castellan):

Gay-Lussac made measurements of volume of a fixed mass of gas under fixed pressure and found a linear relationship

$$V=a+bt$$

where $t$ is temperature.

The vertical axis intercept is $a=V_0$, the volume at $0^{\circ}\text{C}$. The slope is $\left (\partial V/\partial t\right )_p$ and so

$$V=V_0+\left (\partial V/\partial t\right )_pt$$

Charles's experiments showed that the relative increase in volume per degree increase in temperature was the same for all gases on which he made measurements.

That is,

$$\alpha_0=\left ( \frac{\partial V}{\partial t} \right ) \cdot \left ( \frac{1}{V_0} \right )$$

is the same for all gases. This is the coefficient of thermal expansion at $0^{\circ}\text{C}$.

Thus we can write

$$V=V_0(1+\alpha_0t)=V_0\alpha_0\left ( \frac{1}{\alpha_0}+t\right )$$

which expresses the volume of the gas in terms of volume at zero degrees celsius and a constant $\alpha_0$ which is the same for all gases and "as it turns out is very nearly independent of the pressure at which the measurements are made".

If we measure $\alpha_0$ at various pressures we find that fro all gases $\alpha_0$ approaches the same limiting value at $p=0$.

We then make a linear transformation of $t$ coordinates

$$T=\frac{1}{\alpha_0}+t$$

and this defines a new temperature scale called the ideal gas temperature scale.

"The importance of this scale lies in the fact that the limiting value of $\alpha_0$, and consequently $\frac{1}{\alpha_0}$ is the same for all gases. On the other hand, $\alpha_0$ does depend on the sale of temperature originally used for $t$. If $t$ is in degrees celsius then $1/\alpha_0=273.15^{\circ}\text{C}$."

My quip with the above is that is started with a temperature scale (Celsius) that wasn't clearly defined to begin with.

 

[zenterix]

The statement refers to the behavior of the temperature scale. The limit assumes the volume to be fixed when the limit is considered (this is a thought exercise). It is a definition.

How can the limit of $PV$ as $P$ approaches zero with $V$ fixed be constant?

 

[hutchphd]

PV is constant for a fixed temperature for a given system.

 

[zenterix]

PV is constant for a fixed temperature for a given system.

How we we interpret this statement in the context of the following statement

$PV=$ constant for a specified amount of gas at a specified temperature. Experimentally this is strictly true only in the limit of zero pressure.

If we are at zero pressure, then what does this mean for volume if PV=constant? You seemed to say before that in this limit volume is fixed.

The original reasoning started with isotherms, ie a relationship between two independent variables P and V along a curve on which PV is constant. What does it mean to say that this relationship is strictly true in the limit where, as I understand it, pressure is fixed at zero?

 

[hutchphd]

The limit is the value as P approaches zero. Take the isotherm and extrapolate it smoothly to zero P. Obviously if you are on a curve P and V are not independent.

 

[zenterix]

"Take the isotherm and extrapolate it smoothly to zero P."

No idea what this means.

 

[hutchphd]

Do you have an isotherm graph of P vs V?

 

[zenterix]

Do you have an isotherm graph of P vs V?

 

[zenterix]

In the course I am following (5.60 "Thermodynamics and Kinetics" on MIT OCW), in the beginning of one of the lectures (around the 2:50 mark) the lecturer says that the ideal gas theorem is

$$\lim\limits_{p\to 0} (P\bar{V})=constant=f(T)$$

Roughly, what he says (and I am paraphrasing here with the same sort of imprecision that he says it with): take a gas, measure the pressure and the molar volume; change the pressure and measure the molar volume again and so on, getting the pressure smaller and smaller, and the limit turns out to be a constant that is independent of the gas. The constants turns out to be a function of the temperature.

I don't get this. When you "take a gas" and measure the pressure and molar volume, how is this done? I suppose that you have one mole of gas, and implicitly the temperature is constant? If so then you basically change the volume until you get to the pressure you are looking for. Then you change the volume again to get to a lower pressure, and so on. At some point, the pressure becomes zero or close to zero, and you have some volume. Does this mean that in a plot of P vs V there is a finite intercept on the x-axis?

 

[hutchphd]

1. You take a mole of gas and put it into a container of voume V. You measure the pressure (ad temperature if you wish).
2. You lower the temperature (thus lowering the pressure). Use ice or liquid nitrogen as you wish
3. Do this repeatedly

Use the ideal gas equation to solve for T. Experimentally the results will be better for low pressure.