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zenterix

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- 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.

The book says

If pressure is zero, then the lefthand side is zero, so the righthand side must be zero as well, right?

The book goes on

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

How can we even have zero pressure in these equations?

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 trueonly 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 exactonly 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?