Chapter 2   The First Law

2.1   Simple Processes

Isothermal:

constant temperature experiment. This is usually done by maintaining contact with a reservoir (eg. ice bath).

Adiabatic:

there is no heat flow in the experiment across boundaries.

Isobaric:

constant pressure experiment.

Isochoric:

constant volume experiment.

Figure 2.1 - an Indicator ( PV ) Diagram

Figure 2.2 - PV plot of Isotherms

2.2   Work

Figure 2.3 - Work Done By A Gas

2.2.1   An Example

Figure 2.4 - PV Diagram of Work Being Done

2.2.2   Exact and Inexact Differentials

1. if udx+vdy=df , for some function f(x,y) , then the differential is exact

   ó õ

   (udx+vdy)=

   ó õ df=[f]end points

   therefore the differential is independent of path Also we have that

   df=

   ¶ f ¶ x

   dx+

   ¶ f ¶ y

   dy

   Þ u=

   ¶ f ¶ x

   , v=

   ¶ f ¶ y

   this can also be written as (if you differentiate u by y and v by x )

   ¶ v ¶ x

   =

   ¶ u ¶ y

2. if udx+vdy¹ df for any f , then the intergral ò (udx+vdy) depends on the path and so the differential is inexact.

An Exact Differential Example

An Inexact Differential Example

Figure 2.5 - An Inexact Differential

2.2.3   Back to Work Examples

2.3   First Law

First Law of Thermodynamics:

For every adiabatic process between two equilibrium states, work done is the same (they don't need to be QS).

1. adiabatically
2. non-adiabatically (heat flows)

Figure 2.5 - Heat Flowing on a PV Diagram

2.3.1   New Ideas

1. existence of U
2. conservation of energy
3. definition of heat

2.3.2   Conservation of Heat

Figure 2.6 - The Conservation of Heat

2.4   Internal Energy of An Ideal Gas

Figure 2.7 - Joule's Experiment of Free Expansion

• T_initial=T_final     (2.10)
• the first law says
  U_initial=U_final
  ie.
  U(T_initial,V_initial)=U(T_final,V_final)
  U(T,V_initial)=U(T,V_final)
  because T_initial=T_final=T say

  Þ

  æ ç ç è

  ¶ U ¶ V

  ö ÷ ÷ ø

  T

  =0  (for an ideal gas)

  U=U(T)     (2.11)
• another consquence of Joule's experiment
  
  Figure 2.8 - P_initialV_initial=P_finalV_final
  But the process between is a non-static process ( P , T , ... are non-uniform and time dependent). So this inbetween region cannot be plotted on a PV diagram, however Joule's experiment implies that T_initial=T_final so the end-points lie on an isotherm.

2.4.1   specific heat at constant volume

2.4.2   specific heat at constant pressure

• C_p , C_v are functions of T only
• for monotomic gases

  Cv=

  3 2

  Nk,  Cp=

  5 2

  Nk

• for diatomic

  Cv=

  5 2

  Nk,  Cp=

  7 2

  Nk

2.5   Equation of an Adiabat

Figure 2.8 - PV Isoterm Reminder

2.6   Enthalpy

2.7   Some Mathematical Results - See Handout