1. Lecture 7 – The Second Law (Ch. 2)
Wednesday January 23rd
Entropy – a function of state
The Clausius inequality
The combined 1st and 2nd laws
Calculation of entropy for irreversible processes
The TdS equations
Reading: All of chapter 2 (pages )
Homework 2 due on Friday (25th)
Assigned problems, Ch. 2: 6, 8, 16, 18, 20
Homework assignments available on web page
I will post the next assignment on Friday

2. Heat reservoir at temperature T2 > T1 Q2 Q1 Heat reservoir at
Entropy and the 2nd Law
Heat reservoir at
temperature T2 > T1
Efficiency (h): Q2
Heat
Engine Q1
Heat reservoir at
temperature T1 < T2

3. The Clausius Inequality and the 2nd Law
Divide any reversible cycle into a series of thin Carnot cycles, where the isotherms are infinitesimally short: P v
We have proven that the entropy, S, is a state variable, since the integral of the differential entropy (dS = dQ/T), around a closed loop is equal to zero, i.e. the integration of differential entropy is path independent!

4. The Clausius Inequality and the 2nd Law
For an irreversible process
Therefore:
Leading to:
The 2nd expression is the quite general Clausius inequality for which the equality applies only to completely reversible processes.

5. The Clausius Inequality and the 2nd Law
Consider the following cyclic process: P 2
Irreversible
Reversible 1 V
Recalling the definition of entropy:

6. The Clausius Inequality and the 2nd Law
The Clausius equality leads to the following relation between entropy and heat:
This mathematical statement holds true for any process. The equality applies only to reversible processes.
For an isolated system, đQ = 0, therefore
The entropy of an isolated system increases in any irreversible process and is unaltered in any reversible process. This is the principle of increasing entropy.
This leads to the following statement:

7. The combined 1st and 2nd Laws
The 2nd law need not be restricted to reversible processes:
đQ is identifiable with TdS, as is đW with PdV, but only for reversible processes.
However, the last equation is valid quite generally, even for irreversible processes, albeit that the correspondence between đQ & TdS, and đW & PdV, is lost in this case.

8. The Tds equations

9. Entropy associated with irreversible processes
Universe at
temperature T2 Q T1
T2 > T1

11. The TdS equations These equations give:
heat transferred (đQ = TdS) in a reversible process;
the entropy by dividing by T and integrating;
heat flow in terms of measurable quantities;
differences in specific heat capacities, CP, CV, etc..;
relationships between coordinates for isentropic processes.
These equations give: