# For same temperature, mass and composition the pressure and volume are independent coordinates?

From the graph on this I concluded that the pressure and volume are dependent, also it is clearly written that pressure is inversely proportional to volume for fixed mass and temperature.

I was going through the book Heat and thermodynamics by Dittman and Zemansky. The following is from the book: Please see the highlighted region. It is saying that pressure and volume are independent coordinates.

Is the book wrong or writers want to say something else? Kindly explain if the writers want to say something else.

• assuming a diluted gas of 1 mole you have $pV=RT$ so for constant $p$ the volume is proportional to the temperature. – hyportnex Jan 16 '17 at 17:01
• There seems to be a misprint in the book. For constant mass, composition, and temperature, either pressure is an independent variable or volume is an independent variable, but both can't be independent variables at the same time, or the ideal gas law would not be valid. – David White Sep 7 '18 at 1:24

for once it's probably just easier to understand what is meant by looking at the combined gas equation $$pV=kT$$ Boyle didn't really cover the temperature variation, and the dialogue seems to be about that. It's wrong to say that p and V are "independent", as the law clearly constraints them in a relation. I believe what is intended is that boyle's starting point was the treatment of p and V as orthogonal (independent dimensions) - from which his measurements showed that they are dependent.

In my opinion you have discovered an "errata" in the text you cite - the highlighted text and some of the following text does appear make a few incorrect statements. Most large science texts contain a few slight misstatements. Let's begin with some background information to identify what I believe these errata are.

The ideal gas law is derived from the kinetic theory of an ideal gas. An ideal gas is defined as a gas consisting of individual gas particles that have no internal structure. Furthermore, if the particles are in motion, any collisions with other ideal gas particles (either in motion or at rest) or with the inner walls of a container that is confining the particles - are perfectly elastic.

The total kinetic energy of colliding particle is conserved (remains constant) in elastic collisions. It is also assumed that the gas particles never experience any attractive forces to either the inner container walls or to any other ideal gas particles.

Most real gases behave very closely to the behavior predicted by the kinetic gas theory at sufficiently high temperatures and sufficiently low pressures.

The ideal gas equation can describe a "system" consisting of a sealed container - such as a hollow glass bulb or a hollow metallic cylinder - that contains a arbitrary number of moving ideal gas particles. Real gases - within a broad temperature and pressure range - will also be well-described by the ideal gas equation. The equation is shown below.

$$PV={nRT}$$

The pressure "P" is the force per unit area that the gas particles are exerting on the inner walls of the container due to collisions of the particles with the inner container walls, "V" is an arbitrary inner volume of the container. Temperature "T" is related to the kinetic energies (KE) of the gas particles.

The variable "n" is the number of gas particles in molar units (${6.022}$ x $10^{23}$ particles/mole) inside the container and R is a gas constant. If we use total numbers of gas particles as N instead of molar equivalents of particles (n) the ideal gas equation is PV=NkT where k is Boltzman's constant.

Since R (or k) in the ideal gas equation is a constant, and since we assume no increase or decrease in the number of gas particles (so n or N is constant) in the system, we can show the relationship between our ideal gas system from an initial time (or time zero) to a point in time in the future.

$$\frac{(P_i)(V_i)}{T_i}={(R)(n)}=\frac{(P_f)(V_f)}{T_f}$$

The subscript "i" is for P,V,T of the system at time zero (the initial state). The subscript "f" indicates the P,V and T at some arbitrary time in the future (the final state).This is shown by the equation

$$\frac{(P_i)(V_i)}{T_i}=\frac{(P_f)(V_f)}{T_f}$$

Boyle's Law is a special case of this equation where temperature remains constant between initial and final states of the system. Since Ti=Tf the equation simplifies to:

$$(P_i)(V_i)=(P_f)(V_f)$$

For your question, notice that Boyle's law can also be stated as

(P)(V)=constant for any given temperature "T".

So for any given number of gas particles in our system (P)(V) is FIXED. It can not be any other value. You can however - over a large experimental range - choose any volume for the system. There will, however be only one pressure value corresponding to any given volume - namely the pressure that gives the PV constant for that given volume. Conversely, you can choose any pressure within the experimental range you wish but there will be only one corresponding volume (by the same reasoning).

If you graph this PV relationship you get the graph in your link with increasing volume "V" on the x-axis and increasing pressure "P" on the y-axis. Shown this way pressure "depends" on volume but we could interchange P and V on the x and y axis and then volume would "depend" on pressure. The graph's curve shows a range of x-y coordinates or volume-pressure coordinates representing pairs of specific temperature and specific volume values that can exist together for a ideal gas in a system with a given temperature and containing a given number of ideal gas particles.

I believe that the author of the passage you highlighted in your question means to say something like this but unfortunately the author states that in a system with constant number of particles (n or N) and a constant temperature "and if the pressure is kept constant the volume may vary over a wide range of values and vice versa". this is incorrect. The reason it is incorrect is shown by rearranging the ideal gas equation to solve for volume

$$V = \frac{nR}{PT}$$

Since n, T and P are fixed (by the authors statement) and R is a constant, there is only one possible value for V and not a wide range of values for V.

By similar reasoning "vice versa" (if volume "V" is kept constant, "P" may vary over a large range of values) is also wrong. Again, rearrange the ideal gas equation to solve for pressure

$$P=\frac{nR}{VT}$$

Since n, T and V are fixed (by the "vice-versa" in the author's statement) and R is a constant, there is only one possible value for "V" and not a wide range of values.

It is important to acknowledge that simply because there is a (possibly) incorrect statement in a textbook it does not mean that the author actual wrote this. More often the error occurs at a later stage of the publication process - where transcription mistakes frequently occur and are not identified before publication