Thermodynamics

Thermodynamics

Key Point : learn how to describe macroscopic thermal properties of many body systems.

Thermal equilibrium and temperature

macroscopic system : A system consists of many “microscopic” particles or elements (e.g. gases, liquids, solids)

“NOTE

“microscopic” : is relative to “macroscopic”.

“many” : $1 \mathrm{cm}^3$ gas consists of about $10^{19}$ molecules.

Thermodynamics is an empirical theory for the macroscopic systems, typically based on experiments or observations, rather than microscopic details of the system.

A macroscopic state is describes by “a complete set”(完备集) of parameters.

• The number of parameters in the complete set is fixed for a system.
• All other parameters can be expressed as functors of the complete set of parameters.

Classifications of the macroscopic states:

• Equilibrium state(平衡态)

  parameters do not depend on time $t$.

• Non-equilibrium state

  some parameters evolves as time $t$, or more complicated.

Classifications of the systems:

• An isolated system(孤立系统)

  nothing is exchanged with the environment

• A thermally isolated system(绝热系统)

  no “heat” is exchanged with the environment

Classifications of the parameters:

• Extensive parameters: propertional to $V$

• Intensive parameters: independent of $V$

Thermal equilibrium and temperature

Temperature is a thermal observables. How to understand it?

thermal equilibrium : Two system, after a long time touching each other, become equally hot or cold. We call that they are in thermal equilibrium.

The zero-th law : If systems $A$ and $B$ are each in thermal equilibrium with a third system $C$, then $A$ and $B$ are in thermal equilibrium with each other.

The parameters $T$, the temperature, is to describe how hot or cold the system is. And we assume the temperatures of two system in thermal equilibrium are the same.

$T$ is a property of the macroscopic state. So we can describe it with many parameters in a complete set:

$$\begin{align} f(T,x_1,x_2,\cdots,x_M) = 0 \end{align}$$

This is called the equation of state.

“NOTE

E.g.

• The Ideal gas

$$\begin{aligned} PV = NkT \end{aligned}$$

• Real gases

$$\begin{aligned} PV &= A + BP + CP^2 + \cdots\\ &= A + \frac{B'}{V} + \frac{C'}{V^2} + \cdots \end{aligned}$$

Temperature scales

We can set $T=0$ for the transition point of water from liquid to solid, and $T=100$ for that from gas to liquid is. then:

$$\begin{gather} T = aV + b\\ a = \frac{100}{V_f - V_i},\quad b=-\frac{100V_i}{V_f - V_i} \end{gather}$$

Or we can choose a better choice, the ideal gas.

the Boyle’s law : for a given ideal gas at a fixed temperature,

$$\begin{align} PV = \mathrm{constant} \end{align}$$

and the constant depends on the temperature. Therefore, we have

$$\begin{align} T_v = 100\frac{P - P_i}{P_f - P_i},\quad T_P = 100\frac{V - V_i}{V_f - V_i} \end{align}$$

it can be proof that $T_V = T_P$.

The first law

Thermodynamics transformation

A change of state is a Thermodynamics transformation.

quasi-state(准静态) : a transformation with the external condition changing so slowly that at any moment the system is approximately in equilibrium.

“NOTE

In other words, the relaxation(弛豫过程) at each time $t$ from the Non-equilibrium state to the equilibrium one is much faster than the change of the external condition.

“NOTE

If not specified, wo concern ourselves only with the quasi-static transformation in this course.

For a system described by $(P,V)$ , the work done by the system in an infinitesimal transformation is

$$\begin{align} \dbar W = P \dd V \end{align}$$

For a finite process

$$\begin{align} \Delta W = \int \dbar W = \int P \dd V \end{align}$$

“Heat” is what is absorbed by a system, if its temperature increases while no work is done.

We have

$$\begin{align} \dbar Q = C \dd T \end{align}$$

$C$ is called the heat capacity(depends on the detailed nature of the system, process-dependent)

$C$ can not be simply written as $C(P,V)$ ,and an additional direction has to be specified at each point $(P,V)$, such as $C_V(P,V)$ and $C_P(P,V)$.

The first law

For an arbitrary transformation given the initial and final states, the first law states that the quantity $\Delta U$ defined by

$$\begin{align} \Delta U = \Delta Q - \Delta W \end{align}$$

$U$ is called the internal energy. Experimentally, it is found that $U$ is extensive. For an infinitesimal transformation

$$\begin{align} \dd U = \dbar Q - \dbar W \end{align}$$

is exact. That is, $\dd U$ is a differential.

“NOTE

For example, if

$$\begin{aligned} U &= U(P,V)\\ \dd U &= F_P \dd P + F_V \dd V \end{aligned}$$

leads to

$$\begin{aligned} \frac{\partial F_P}{\partial V} = \frac{\partial F_V}{\partial P} \end{aligned}$$

Some applications

1. Exercises:

   $$\begin{align} \dbar Q &= \left(\frac{\partial U}{\partial P}\right)_V \dd P + \left[\left(\frac{\partial U}{\partial V}\right)_P + P\right] \dd V\\ \dbar Q &= \left[\left(\frac{\partial U}{\partial T}\right)_P + P\left(\frac{\partial V}{\partial T}\right)_P\right] \dd T + \left[\left(\frac{\partial U}{\partial P}\right)_T + P\left(\frac{\partial V}{\partial P}\right)_T\right] \dd P\\ \dbar Q &= \left(\frac{\partial U}{\partial T}\right)_V \dd T + \left[\left(\frac{\partial U}{\partial V}\right)_T + P\right] \dd V \end{align}$$

   These are called the heat equations. It can be deduced from

   $$\begin{align} C_V &= \left(\frac{\partial Q}{\partial T}\right)_V = \left(\frac{\partial U}{\partial T}\right)_V\\ C_P &= \left(\frac{\partial Q}{\partial T}\right)_P = \left(\frac{\partial H}{\partial T}\right)_P \end{align}$$

   $H = U + PV$ called enthalpy(焓).

2. Free expansion of an ideal gas

   Since $\Delta W = 0$, $\Delta T = 0$ Thus $\Delta Q = 0$, $\Delta U = 0$

3. Internal energy of an ideal gas

   Since $U$ depends only on $T$,

   $$\begin{align} C_V = \left(\frac{\dd U}{\dd T}\right)_V \end{align}$$

   Assuming $C_V$ to be independent of $T$, we obtain

   $$\begin{align} U = C_V T + \mathrm{constant} \end{align}$$

4. $C_P - C_V$ for an ideal gas

The second law

Reversible and irreversible transformations

A reversible transformation(可逆过程) is a transformation that the system retraces its history in time when the external condition retraces its history in time.

The second law

• Kelvin statement : There exits no Thermodynamics transformation whose sole effect is to extract a quantity of heat from a given heat reservoir and to convert it entirely into work.

• Clausius statement : There exits no Thermodynamics transformation whose sole effect is to transfer a quantity of heat from a colder reservoir to a hotter reservoir.

“NOTE

Adiabatic(绝热的) expansions don’t violate the second law, since the work is not converted from heat, rather from the change of state.

Kelvin and Clausius statements are equivalent.

Carnot’s theorem

the Carnot engine

An engine(热机) is a machine which convert heat into work. An engine which does everything in a reversible way is called a Carnot engine(卡诺热机).