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Thus, when we say two objects (a thermodynamic system and its environment, for example) are in thermal equilibrium , we mean that they are at the same temperature, as we discussed in Temperature and Heat . Let us consider three objects at temperatures ${T}_{1},\phantom{\rule{0.2em}{0ex}}{T}_{2},$ and ${T}_{3},$ respectively. How do we know whether they are in thermal equilibrium? The governing principle here is the zeroth law of thermodynamics , as described in Temperature and Heat on temperature and heat:
If object 1 is in thermal equilibrium with objects 2 and 3, respectively, then objects 2 and 3 must also be in thermal equilibrium.
Mathematically, we can simply write the zeroth law of thermodynamics as
This is the most fundamental way of defining temperature: Two objects must be at the same temperature thermodynamically if the net heat transfer between them is zero when they are put in thermal contact and have reached a thermal equilibrium.
The zeroth law of thermodynamics is equally applicable to the different parts of a closed system and requires that the temperature everywhere inside the system be the same if the system has reached a thermal equilibrium. To simplify our discussion, we assume the system is uniform with only one type of material—for example, water in a tank. The measurable properties of the system at least include its volume, pressure, and temperature. The range of specific relevant variables depends upon the system. For example, for a stretched rubber band, the relevant variables would be length, tension, and temperature. The relationship between these three basic properties of the system is called the equation of state of the system and is written symbolically for a closed system as
where V , p , and T are the volume, pressure, and temperature of the system at a given condition.
In principle, this equation of state exists for any thermodynamic system but is not always readily available. The forms of $f(p,V,T)=0$ for many materials have been determined either experimentally or theoretically. In the preceding chapter, we saw an example of an equation of state for an ideal gas, $f(p,V,T)=pV-nRT=0.$
We have so far introduced several physical properties that are relevant to the thermodynamics of a thermodynamic system, such as its volume, pressure, and temperature. We can separate these quantities into two generic categories. The quantity associated with an amount of matter is an extensive variable , such as the volume and the number of moles. The other properties of a system are intensive variable s , such as the pressure and temperature. An extensive variable doubles its value if the amount of matter in the system doubles, provided all the intensive variables remain the same. For example, the volume or total energy of the system doubles if we double the amount of matter in the system while holding the temperature and pressure of the system unchanged.
Consider these scenarios and state whether work is done by the system on the environment (SE) or by the environment on the system (ES): (a) opening a carbonated beverage; (b) filling a flat tire; (c) a sealed empty gas can expands on a hot day, bowing out the walls.
a. SE; b. ES; c. ES
A gas follows $pV=bp+{c}_{T}$ on an isothermal curve, where p is the pressure, V is the volume, b is a constant, and c is a function of temperature. Show that a temperature scale under an isochoric process can be established with this gas and is identical to that of an ideal gas.
$p\left(V-b\right)=\text{\u2212}{c}_{T}$ is the temperature scale desired and mirrors the ideal gas if under constant volume.
A mole of gas has isobaric expansion coefficient $dV\text{/}dT=R\text{/}p$ and isochoric pressure-temperature coefficient $dp\text{/}dT=p\text{/}T$ . Find the equation of state of the gas.
Find the equation of state of a solid that has an isobaric expansion coefficient $dV\text{/}dT=2cT-bp$ and an isothermal pressure-volume coefficient $dV\text{/}dp=\text{\u2212}bT.$
$V-bpT+c{T}^{2}=0$
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