Zeroth Law of Thermodynamics

[Kakashi]

Consider three systems A,B,and C each containing a fixed amount of gas, whose equilibrium state is specified by two independent intensive variables. We have only a qualitative notion of temperature through our senses, and the zeroth law allows us to define temperature quantitatively.

Assume A,B, and C are each initially in equilibrium, so their intensive variables are uniform in space and constant in time. If we bring them into thermal contact, heat transfer will occur. Suppose this heat transfer takes place very slowly. During the process the systems may not be in equilibrium, but eventually A,B, and are in thermal equilibrium with one another. We associate some measurable property (such as color, electrical conductivity, or expansion) with temperature in order to signal when thermal equilibrium has been reached and whether temperature is changing or remaining constant.

Once thermal equilibrium is established, the systems are at the same temperature. If we now change the pressure of one system while maintaining thermal equilibrium, only one additional intensive variable is needed to specify the state. The curve of all pressure–volume pairs at fixed temperature is an isotherm.My confusion is the following: what is the purpose of bringing A,B, and C into thermal contact in the first place? Is it simply to fix the temperature so that the relationship between pressure and volume can be studied? Could we not instead insulate the system and directly carry out Boyle’s law experiments?

My textbook then relates this discussion to Boyle’s law. Historically, Boyle studied compressed air at (approximately) constant temperature using a mercury column in a U-tube. When Boyle plotted the reciprocal of the pressure versus the volume of trapped air, the data lay close to a straight line with an intercept near zero. However, the book states that the relation$$ PV=Constant $$ is only strictly valid in the limit of zero pressure.Where does this “limit of zero pressure” come from?

As I understand it, if the volume is increased slowly while maintaining thermal contact, the temperature remains constant and the pressure decreases. At sufficiently small pressures, the plot of volume versus 1/P approaches a straight line, and this behavior is independent of the gas used (for a fixed amount of gas). Hence the constant in Boyle’s law depends only on temperature.This motivates writing

$$ PV=f(T) $$ where f(T) is an unspecified function that depends on T.

Why is it then justified to choose this function to be linear and write$$ \frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}=nR $$

Finally, the book defines temperature as $$ T=\frac{\lim_{P \to 0} (P \bar{V})}{R} $$

How should this definition be interpreted? Is temperature being treated as a function of both pressure and volume, or is one variable held fixed while the other varies? At finite pressure, T≠PV/R but does PV/R approach the temperature as the pressure is reduced toward zero?

 

[anuttarasammyak]

I just comment on limit $$P \rightarrow 0$$. As P has physical dimension it seems inappropriate to take its limit. It could be $P$/ [constant which has physical dimension of pressure] << 1.
And
$$\frac{\lim_{P \to 0} (P \bar{V})}{R}=0$$
How does your textbook define taking the limit ?

 

[Borek]

Where does this “limit of zero pressure” come from?

What it tries to say is that the relationship holds precisely only for ideal gas (pointlike molecules with zero volume, no interactions other than elastic collisions), and that the lower the pressure, the closer the real gas is to the ideal gas.

This is quite common in the real systems, you will see the same approach in liquid mixtures/solutions - many of their properties depend on the concentration of the solute, but they are perfectly linear in concentration only for "infinitely diluted" solution.

 

[Kakashi]

I just comment on limit $$P \rightarrow 0$$. As P has physical dimension it seems inappropriate to take its limit. It could be $P$/ [constant which has physical dimension of pressure] << 1.
And
$$\frac{\lim_{P \to 0} (P \bar{V})}{R}=0$$
How does your text define taking the limit ?

PV/R is not equal to 0 because the molar volume is not fixed. As the pressure decreases the molar volume increases and $$ \frac{P\bar{V}}{R} $$ approaches T.

 

[Kakashi]

What it tries to say is that the relationship holds precisely only for ideal gas (pointlike molecules with zero volume, no interactions other than elastic collisions), and that the lower the pressure, the closer the real gas is to the ideal gas.

This is quite common in the real systems, you will see the same approach in liquid mixtures/solutions - many of their properties depend on the concentration of the solute, but they are perfectly linear in concentration only for "infinitely diluted" solution.

So do we make this assumption because a gas approaching zero pressure corresponds to increasing the volume without bound, so that intermolecular interactions become negligible while the system remains in thermal equilibrium, and this allows temperature to be defined independently of the substance? Also, does this definition imply that temperature cannot be measured for systems with finite volume?

 

[anuttarasammyak]

PV/R is not equal to 0 because the molar volume is not fixed. As the pressure decreases the molar volume increases and PV¯R approaches T.

I got $\bar{V}$ is molar volume. Thanks.

A gas with a known number of moles n is placed in a container whose volume can be varied and is brought into thermal equilibrium with a heat reservoir. In the limit where the volume is increased while maintaining equilibrium, the temperature of the heat reservoir can be determined from the equation
$$T=\frac{PV}{nR}$$
This is my interpretation of the textbook.　Assuming that the gas pressure is initially in equilibrium with atmospheric pressure, if the air pressure in a closed laboratory is reduced and approaches a vacuum, this can be regarded as taking the limit in which the pressure is made smaller and smaller.

Also, does this definition imply that temperature cannot be measured for systems with finite volume?

It is, in principle, possible to define temperature based on the Boyle–Charles law (Charles’s law); however, as a practical thermometer, this approach has problems in terms of accuracy and convenience and is therefore not suitable for practical use.