**Example 1**

Consider a piston like the ones discussed previously with a radius of 8 cm. The pressure inside the pistons is about equal to the atmospheric pressure, and, due to some internal reactions, the system expands by about 5 cm. Find the volume change and the work done by the system on the surrounding.

**Solution 1**

Start by converting to SI. Next, set up your equation to solve the problem.

*r* = 8 cm = 0.08 m

ΔL = 5 cm = 0.05 m

P = 1.013 · 10^{5}

Δ*V* = ?

*W*= ?

The change in volume, for a cylinder, is the area of the section, a circle, multiplied by the height.

Δ *V* = *A* · Δ*L*

*A* = π · *r*^{2}

Δ *V* = π · *r*^{2} · Δ*L* = π · (0.08 m)^{2} · 0.05 m

Δ*V* = 1 · 10^{–3} m^{3}

*W* = –P · Δ*V* = –1.013 · 10^{5} · 1 · 10 ^{–3} m^{3}

*W* = –1.013 · 10^{2}N · m

or with one number of significant figures, as the problem gives us: *W* = – 1.10^{2} *J*.

The work is negative because the system makes work on its surrounding, or, in other words, is losing some of its energy.

A geometric approach to interpreting the formula for work offers a second method to measure work performed in a thermodynamic process. Because work is proportional to pressure and volume, we can represent the two parameters and graph their dependence. In the previous case where volume is increasing, the graph will look like the one in Figure 12.2.

The arrow shows us the direction of the process (the volume is increasing; there is work done by the system on its surroundings to push the air out and make room for itself). The definition of work also considers the product of the pressure and the change in volume. According to the geometry of the graph, this is exactly the area darkened between the initial and final volumes and the constant pressure.

The graph is called a *PV-diagram*, and the work can be calculated by finding the area under the graph in the PV-diagram.

**Example 2**

Consider the graph in Figure 12.2. Reading the measurements from the graph, calculate the amount of work performed by the system on the surroundings.

**Solution 2**

To our advantage, all data is already in SI. The area is a simple geometrical shape: a rectangle. The area under the graph will be:

*width* · *height* = (201 · 10^{–3}m^{3}) · 1· 10^{5} N/m^{2} = 1 · 10^{2} · J

The decision regarding the sign will have to come from interpreting the graph according to the convention: Expanding objects yield a negative value of work. So, our final result is:

*W* = –1 · 10^{2}J

This is exactly the same as the previous answer, with one error. The error stems from the fact that the graph does not give enough detail on the pressure measurements.

**Zeroth Law of Thermodynamics**

We have seen that pressure and volume are mechanical parameters. There are also other properties of a system that are dependent on the motion of the constituent atoms and molecules. Characterizing this motion is not an easy job though, because the number of particles in anyone material is astronomic. In one

mole of every substance, you can count a certain number of particles; that number is called Avogadro's number, or *N*_{A} = 6.022 . 10^{23} particles/mol. Given the number of particles in a substance, N, one can find the number of moles from:

*n* = *N*/*N*_{A}

Also, the mass of the substance can be determined based on the number of particles and the mass per particle:

*m* = *m*_{particle} · *N*

With this, we can define the parameters necessary to completely characterize a thermodynamic state: pressure, *P*, volume, *V*, number of moles, *n*, and mass, *m*.

A first law that summarizes one of these parameters, temperature, is the *zeroth law of thermodynamics*. In all natural phenomena, two or more systems are in thermal contact. They can exchange energy through heat and, after a period of time, will have the same temperature: The colder object warms up and the warmer object cools down. In other words, they will ultimately be in *thermal equilibrium*. To generalize this behavior, the zeroth law states: Two systems, each in thermal equilibrium with a third system (1 with 2, and 2 with 3), are in thermal equilibrium with each other (l with 3).

Hence, each part of this system has the same temperature. The idea of equilibrium is so important that it was determined that it was necessary for this statement to preceede the first and second laws of thermodynamics, thus the name *zeroth*. The contact between two or more objects when they exchange heat but no mass is called *thermal contact*.

Zeroth Law of Thermodynamics

Two systems, each in thermal equilibrium with a third system (1 with 2, and 2 with 3), are in thermal equilibrium with each other (1 with 3).

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