2 The first law 2.3 Specific heats 2.5 Energy

2.4 The pressure

(September 1, 2023)

Generally the pressure is a function of temperature $T$ , volume $V$ , and number $N$ , $p(T,V,N)$ . Usually the dependence on $N$ is not notated, as $N$ is considered to be a fixed constant, i.e. $p(T,V)\equiv p(T,V,N)$ .

The volume per particle is the inverse of the density

v_{N}\equiv\frac{V}{N}=\frac{1}{n}

(2.21)

The pressure is intensive. This means that if I consider twice as many particles at the same temperature and desity the pressure is unchanged.

Since the pressure is intensive, $p=p(T,V,N)$ is only a function of temperature and $v_{N}=V/N$ , and not $V$ and $N$ separately, $p=p(T,v_{N})$ . If you prefer, you can parametrize the pressure by the temperature and density, $p=p(T,n)$ . Thus, we have three parametrizations of the same physical quantity⁴⁴ 4 We are using a common notation in physics, which mathematicians don’t like. What we are really talking about here is three separate functions (or maps) which return the same value at corresponding arguments $p=p_{1}(T,V,N)=p_{2}(T,v_{N})=p_{3}(T,n)$ (2.22) $p_{1}$ and $p_{2}$ and $p_{3}$ describe the same quantity but have different functional forms, since they are functions of different variables. Take the ideal gas law: $p_{1}(T,V,N)=\frac{N{kT}}{V}\qquad p_{2}(T,v_{N})=\frac{{kT}}{v_{N}}\qquad p_{% 3}(T,n)=n\,{kT}$ (2.23) For a mathematician the map or functional form is paramount, and the name of the argument is irrelevant, e.g. : $p_{3}(T,x,y)=\frac{y}{x}\,{kT}\qquad p_{2}(T,x)=\frac{{kT}}{x}\qquad p_{3}(T,x% )=x\,{kT}$ (2.24) Mathematicians kind of have point: $1/x$ and $x$ are not the same function! The physics notation uses the names of the arguments, $p(T,V,N)$ , $p(T,v_{N})$ , and $p(T,n)$ , to distinguish the functions $p_{1}$ , $p_{2}$ , and $p_{3}$ . The physics notation prevents an explosion of symbols for the same physical quantity, but if confused you should go back to the math notation. :

p=p(T,V,N)=p(T,v_{N})=p(T,n)

(2.25)

Using the chain rule, we can relate the volume derivatives of $p(T,V,N)$ to derivatives with respect to the volume-per-particle or with respect to the density :

\left(\frac{\partial p}{\partial V}\right)_{T,N}=\frac{1}{N}\left(\frac{% \partial p}{\partial{v_{N}}}\right)_{T}\qquad\mbox{and}\qquad\left(\frac{% \partial p}{\partial V}\right)_{T,N}=-\frac{N}{V^{2}}\left(\frac{\partial p}{% \partial n}\right)_{T}\,,

(2.26)

Thus, the derivative with respect to volume records how the pressure changes with the particle density⁵⁵ 5 More explicitly we would write $\left(\frac{\partial p}{\partial V}\right)_{T}\equiv\left(\frac{\partial p}{% \partial V}\right)_{T,N}$ :

N\left(\frac{\partial p}{\partial V}\right)_{T}=\left(\frac{\partial p}{% \partial{v_{N}}}\right)_{T}=-n^{2}\left(\frac{\partial p}{\partial n}\right)_{T}

(2.27)

Hopefully this is clear enough.

2.4.1 Pressure, Volume, and the Equation of State

Virial Expansion: First consider gasses. At low density, we can make an expansion in the density $N/V$

p(T,V,N)=\frac{N{kT}}{V}\left(1+B(T)\frac{N}{V}+C(T)\left(\frac{N}{V}\right)^{% 2}+\ldots\right)

(2.28)

At very low density the ideal gas $N{kT}/V$ is valid, while at higher density there are corrections. The $B(T)$ is known first virial coefficient. It corrects the ideal gas pressure by an amount of order $\delta p=B(T){kT}n^{2}$ . This goes like the square of the number of particles, and hence reflects the interactions between them.

Response coefficients: Now consider a general substance. Instead of working with the pressure as a function of temperature and volume, we will often work with the volume as a function of temperature and pressure, which contains the same information. The differential in volume is

dV=\left(\frac{\partial V}{\partial T}\right)_{p}dT+\left(\frac{\partial V}{% \partial p}\right)_{T}dp

(2.29)

The two derivatives characterize the response of the system. The first one characterizes the expansion of the system with increase in temperature

\beta_{p}=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}\equiv\mbox% {Thermal expansion coefficient}

(2.30)

The second one characterizes the increase in pressure with decrease in volume

\kappa_{T}=\frac{-1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}\equiv% \mbox{Isothermal compressiblity}

(2.31)

The derivatives of like $(\partial V/\partial T)_{p}$ are divided by $V$ so that response coefficients, $\beta_{p}$ and $\kappa_{T}$ , are intensive. This means that as the number of particles is increased at fixed pressure and temperature, $\beta_{p}$ and $\kappa_{T}$ are unchanged. Both $(\partial V/\partial T)_{p}$ and $V(T,p)$ are proportional to the number of particles, but $\beta_{p}$ , which is the ratio of these two quantities, is independent of the total number of particles.

The speed of sound and the compressibility: The first coefficient $\beta_{p}$ has a clear everyday meaning, i.e. how much does something expand when heated. The second coefficient $\kappa_{T}$ is inversely related to the stiffness of the material. The isothermal bulk modulus directly reflects the stiffness

B_{T}=-\frac{1}{V}\left(\frac{\partial p}{\partial V}\right)_{T}=\frac{1}{% \kappa_{T}}

(2.32)

At constant temperature, an increase in volume ${\rm d}V$ leads to a drop in pressure determined by $\kappa_{T}$ :

{\rm d}p=-B_{T}\frac{{\rm d}V}{V}

(2.33)

Since sound is a pressure wave, it is not surprising then that $\kappa_{T}$ is related to the speed of sound. The speed of sound is

c_{s}=\sqrt{\frac{\gamma}{\kappa_{T}\rho}}

(2.34)

where $\rho$ is the mass per unit volume, and $\gamma=C_{p}/C_{V}$ is the adiabtic index.

The adiabatic compressibility and sound: The adiabatic compressibility $\kappa_{S}$ is directly related to $\kappa_{T}$ . $\kappa_{S}$ is the decrease in volume with increasing pressure, with no heat flow⁶⁶ 6 The “S” means at fixed entropy, and as the course progresses we will write $\kappa_{S}\equiv-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{S}\,.$ For an ideal gas, an adiabatic expansion means that $pV^{\gamma}={\rm const}$ (see below), which can be used to prove Eq. (2.37) for the simple case of an ideal gas. Proving Eq. (2.37) in general requires a more extensive discussion of entropy, covered later in the course :

\kappa_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{adiab}

(2.35)

Similarly there is an adiabatic bulk modulus

B_{S}=-V\left(\frac{\partial p}{\partial V}\right)_{adiab}=\frac{1}{\kappa_{S}}

(2.36)

We will show much later in the course that for any homogeneous substance

\kappa_{S}=\frac{\kappa_{T}}{\gamma}

(2.37)

and so the speed of sound is naturally expressed using the adiabatic compressibility

c_{s}=\sqrt{\frac{B_{S}}{\rho}}

(2.38)