Thermal & Statistical Physics mbcx8ph2

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Thermal & Statistical Physics

Zeroth Law of Thermodynamics

A Logic Law that just says if A = B and A = C then B = C

3rd Law of Thermodynamics

As a system approaches absolute zero, all processes cease and the entropy of the system approaches a minimum value.

2nd Law of Thermodynamics

Infinitesimal change in entropy for a general process

Change in Entropy for a reversible process, note the equality.

Second Law of Thermdynamics

Statement of the law:

In a system, a process that occurs within a closed system will tend to increase the entropy of the system and entropy can never decrease.

The statement is an equality if the process is reversible.

1st Law of Thermodynamics, Heat and Gases

Useful Relations for Gases

For a reversible aciabatic process on an ideal gas, this holds true for K = const.

Heat Capacity relation for an ideal gas

Heat Capacities

At constant pressure (for a gas)

At constant volume (for a gas)

Types of Gases

Van-der-Waals Gas

Van der Waals argued that the pressure, P_attraction, is the result of mutual attraction between bulk of gas and a sample near the boundary, and hence should be proportional to the product of their densities.

Hard-Sphere Gas

This approximation treats the atoms as hard-spheres. Therefore, it is necessary to exclude the volume of the atoms.

Ideal Gas

Work done for various processes.

On a magnetic material by an external field.

On a wire to stretch it.

On a gas to expand/compress it.

1st Law of Thermodynamics

In the equation I have replaced d-bar with delta's, hopefully it won't confuse people.

Change of Energy = Work done on the system + heat absorbed by the system

The first law is a statement of energy conservation

Open Systems

Phase Equilibrium conditions

Phase Equilibrium occurs when all intensive variables of the two phases are equal. This can be proved by the second law of thermodynamics and by considering the multi-phase flow as a closed system.

Fundamental Thermodynamic Relation

This is for a single-particle species system only. There is a subsequent extension to this relation for multi-particle species system, however I don't think we need to know it.

Statistical Theory Of Thermodynamics

Macrostates and Microstates

The macrostate of a given thermodynamic system is give by simply a few thermodynamic variables, though like, P, V, T and state functions such as E and S. In other words, it's the collective behaviour of the substance.

A microstate of a thermodynamic system is the total states of each individual particle, therefore giving a total description of the gas in question.

Quantum Mechanics

In a qunatum mechanical description, particles are described by wavefunctions which are specified by the set of quantum numbers, n l, m_l, and small_omega (the spin).

For N quantum particles in a box, each state is a plane wave solution, and hence we can use three discrete components of momenta. We use these collection of quantum numbers to form the microstate. This is known as the independant particle approximation, as it ignores the interactions between particles.

Classical Mechanics

In a classical mechanics description, a particle can be described entirely by it's position and momentum. So for N particles, since we have 3 spatial dimensions and 3 momenta dimensions for each particle, the combination of these 6 dimensions for each particle make up what's known as phase space, which is a 6N dimensional space.

Boltzmann's Assumption of entropy

Omega is the number of microstates in the given system.

Maxwell Relations

Clairaut's Therorem (it's the offiical name, don't need to know the name)

Total differential expansion of a general function of n variables

Using this equation, we can expand all of the thermodynamic potentials in terms of their total differential components, and simply compare the coefficients with the ones defined from the fundamental thermodynamic relation to yield a pair of equations that must be true.

Enthalpy

At constant pressure, the change in heat is equal to the change in enthalpy. This value is known as the latent heat.

4th Maxwell Relation

Gibbs Free Energy

This definition is useful for phase transitions, since the Gibbs free energy is always constant at this point (ie, when T and P are fixed).

3rd Maxwell Relation

Helmholtz Free Energy

This definition is useful for systems of constant temperature.

Differential Form

To find the differential forms, differentiate with respect to a random variable, say x for this example, assuming all of the functions are functions of x. Then multiply by dx to eliminate the random variable.

2nd Maxwell Relation

To derive one of the Maxwell equations, follow the following steps:

Fundamental Thermodynamic Relation and natural variables.

The "natural" variables of E are S and V, as the infinitesimal change of E depends on the changes of these two variables.

1st Maxwell Relation

Heat Engine and Cycles

First Law for any closed cycle, C.

Carnot Engines

The Carnot Cycle is as follows:

•

• Reversible isothermal expansion of the gas at the "hot" temperature, T1 (isothermal heat addition). (1 to 2 in image.)

• Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). (2 to 3 in image.)

• Reversible isothermal compression of the gas at the "cold" temperature, T2. (isothermal heat rejection). (3 to 4 in image.)

• Isentropic compression of the gas (isentropic work input). (4 to 1 in image.)

Information taken from:

http://en.wikipedia.org/wiki/Carnot_cycle

Carnot Engine efficiency. (See notes)

The carnot efficiency is independant of the substance used, therefore it can be derived by considering the process occuring through an ideal gas.

Carnot relation.

This can be obtained via the proof of Carnot's Theorem.

Carnot's Theorem: A Carnot engine is the most efficient heat engine.

Efficiency is defined by the following equation.