In_II this_DD1 chapter_NN1 ,_, we_PPIS2 quickly_RR review_VV0 some_DD basic_JJ definitions_NN2 and_CC concepts_NN2 from_II thermodynamics_NN1 ._. 
We_PPIS2 then_RT provide_VV0 a_AT1 brief_JJ description_NN1 of_IO the_AT first_MD and_CC second_MD laws_NN2 of_IO thermodynamics_NN1 ._. 
Next_MD ,_, we_PPIS2 discuss_VV0 the_AT mathematical_JJ consequences_NN2 of_IO these_DD2 laws_NN2 and_CC cover_VV0 some_DD relevant_JJ theorems_NN2 in_II multivariate_JJ calculus_NN1 ._. 
Finally_RR ,_, free_JJ energies_NN2 and_CC their_APPGE importance_NN1 are_VBR introduced_VVN ._. 
A_AT1 state_NN1 function_NN1 is_VBZ a_AT1 function_NN1 that_CST depends_VVZ only_RR on_II the_AT current_JJ properties_NN2 of_IO the_AT system_NN1 and_CC not_XX on_II the_AT history_NN1 of_IO the_AT system_NN1 ._. 
Examples_NN2 of_IO state_NN1 functions_NN2 include_VV0 density_NN1 ,_, temperature_NN1 ,_, and_CC pressure_NN1 ._. 
A_AT1 path_NN1 function_NN1 is_VBZ a_AT1 function_NN1 that_CST depends_VVZ on_II the_AT history_NN1 of_IO the_AT system_NN1 ._. 
Examples_NN2 of_IO path_NN1 functions_NN2 include_VV0 work_NN1 and_CC heat_NN1 ._. 
An_AT1 extensive_JJ property_NN1 is_VBZ a_AT1 characteristic_NN1 of_IO a_AT1 system_NN1 that_CST is_VBZ proportional_JJ to_II the_AT size_NN1 of_IO the_AT system_NN1 ._. 
That_REX21 is_REX22 ,_, if_CS we_PPIS2 double_VV0 the_AT size_NN1 of_IO the_AT system_NN1 ,_, then_RT the_AT value_NN1 of_IO an_AT1 extensive_JJ property_NN1 would_VM also_RR double_VVI ._. 
Examples_NN2 of_IO extensive_JJ properties_NN2 include_VV0 total_JJ volume_NN1 ,_, total_JJ mass_NN1 ,_, total_JJ internal_JJ energy_NN1 ,_, etc._RA extensive_JJ properties_NN2 will_VM be_VBI underlined_VVN ._. 
For_REX21 example_REX22 ,_, the_AT total_JJ entropy_NN1 of_IO the_AT system_NN1 ,_, which_DDQ is_VBZ an_AT1 extensive_JJ property_NN1 ,_, will_VM be_VBI denoted_VVN as_CSA S._NP1 An_AT1 intensive_JJ property_NN1 is_VBZ a_AT1 characteristic_NN1 of_IO a_AT1 system_NN1 that_CST does_VDZ not_XX depend_VVI on_II the_AT size_NN1 of_IO the_AT system_NN1 ._. 
That_REX21 is_REX22 ,_, doubling_VVG the_AT size_NN1 of_IO the_AT system_NN1 leave_VV0 the_AT value_NN1 of_IO an_AT1 intensive_JJ property_NN1 unchanged_JJ ._. 
Examples_NN2 of_IO intensive_JJ properties_NN2 are_VBR pressure_NN1 ,_, temperature_NN1 ,_, density_NN1 ,_, molar_JJ volume_NN1 ,_, etc_RA ._. 
By_II definition_NN1 ,_, an_AT1 intensive_JJ property_NN1 can_VM only_RR be_VBI a_AT1 function_NN1 of_IO other_JJ intensive_JJ properties_NN2 ._. 
It_PPH1 can_VM not_XX be_VBI a_AT1 function_NN1 of_IO properties_NN2 that_CST are_VBR extensive_JJ because_CS it_PPH1 would_VM then_RT depend_VVI on_II the_AT size_NN1 of_IO the_AT system_NN1 ._. 
The_AT first_MD law_NN1 of_IO thermodynamics_NN1 is_VBZ simply_RR a_AT1 statement_NN1 of_IO the_AT conservation_NN1 of_IO energy_NN1 ._. 
Energy_NN1 can_VM take_VVI on_RP a_AT1 variety_NN1 of_IO forms_NN2 ,_, for_REX21 example_REX22 kinetic_JJ energy_NN1 ,_, chemical_JJ energy_NN1 ,_, or_CC thermal_JJ energy_NN1 ._. 
These_DD2 different_JJ forms_NN2 of_IO energy_NN1 can_VM transform_VVI from_II one_PN1 to_II another_DD1 ,_, however_RR ,_, the_AT sum_NN1 total_NN1 of_IO all_DB the_AT types_NN2 of_IO energy_NN1 must_VM remain_VVI constant_JJ ._. 
The_AT second_MD law_NN1 of_IO thermodynamics_NN1 formalizes_VVZ the_AT observation_NN1 that_CST heat_NN1 is_VBZ spontaneously_RR transferred_VVN only_RR from_II higher_JJR temperatures_NN2 to_TO lower_VVI temperatures_NN2 ._. 
From_II this_DD1 observation_NN1 ,_, one_PN1 can_VM deduce_VVI the_AT existence_NN1 of_IO a_AT1 state_NN1 function_NN1 of_IO a_AT1 system_NN1 ._. 
Note_VV0 that_CST the_AT second_MD law_NN1 of_IO thermodynamics_NN1 is_VBZ unique_JJ among_II the_AT various_JJ laws_NN2 of_IO nature_NN1 in_II that_DD1 it_PPH1 is_VBZ not_XX symmetric_JJ in_II time_NNT1 ._. 
It_PPH1 sets_VVZ a_AT1 direction_NN1 in_II time_NNT1 ,_, and_CC consequently_RR there_EX is_VBZ a_AT1 distinction_NN1 between_II running_VVG forward_RL in_II time_NNT1 and_CC running_VVG backwards_RL in_II time_NNT1 ._. 
We_PPIS2 can_VM notice_VVI that_CST a_AT1 film_NN1 is_VBZ being_VBG palyed_VVN in_II reverse_NN1 because_CS we_PPIS2 observe_VV0 events_NN2 that_CST seem_VV0 to_TO violate_VVI the_AT second_MD law_NN1 ._. 
Note_VV0 that_CST in_II an_AT1 isolated_JJ system_NN1 ,_, every_AT1 spontaneous_JJ event_NN1 that_CST occurs_VVZ always_RR increases_VVZ the_AT total_JJ entropy_NN1 ._. 
Therefore_RR ,_, at_II equilibrium_NN1 ,_, where_CS the_AT properties_NN2 of_IO a_AT1 system_NN1 no_RR21 longer_RR22 change_VV0 ,_, the_AT entropy_NN1 of_IO the_AT system_NN1 will_VM be_VBI maximized_VVN ._. 
The_AT Gibbs_NP1 free_JJ energy_NN1 is_VBZ minimized_VVN for_IF a_AT1 system_NN1 for_IF a_AT1 system_NN1 at_II constant_JJ temperature_NN1 ,_, pressure_NN1 ,_, and_CC total_JJ number_NN1 of_IO moles_NN2 ._. 
The_AT Gibbs_NP1 free_JJ energy_NN1 is_VBZ important_JJ because_CS in_II most_DAT experiments_NN2 ,_, the_AT temperature_NN1 and_CC pressure_NN1 are_VBR variables_NN2 that_CST we_PPIS2 control_VV0 ._. 
This_DD1 will_VM become_VVI useful_JJ to_II us_PPIO2 later_RRR when_CS we_PPIS2 consider_VV0 phase_NN1 equilibria_NN2 ._. 
As_CSA we_PPIS2 have_VH0 seen_VVN ,_, free_JJ energies_NN2 such_II21 as_II22 the_AT internal_JJ energy_NN1 and_CC Gibbs_NP1 free_JJ energy_NN1 are_VBR useful_JJ in_CS21 that_CS22 they_PPHS2 tell_VV0 us_PPIO2 whether_CSW a_AT1 process_NN1 will_VM occur_VVI spontaneously_RR or_CC not_XX ._. 
A_AT1 process_NN1 in_II which_DDQ the_AT requisite_JJ free_JJ energy_NN1 decreases_VVZ will_VM occur_VVI spontaneously_RR ._. 
A_AT1 process_NN1 in_II which_DDQ the_AT free_JJ energy_NN1 increases_NN2 will_VM not_XX occur_VVI spontaneously_RR ._. 
This_DD1 does_VDZ not_XX mean_VVI that_CST the_AT process_NN1 can_VM not_XX happen_VVI ;_; we_PPIS2 can_VM force_VVI the_AT process_NN1 to_TO occur_VVI by_II performing_VVG work_NN1 on_II the_AT system_NN1 ._. 
Therefore_RR ,_, we_PPIS2 see_VV0 that_CST free_JJ energies_NN2 are_VBR useful_JJ to_II us_PPIO2 ,_, qualitatively_RR ,_, in_CS21 that_CS22 they_PPHS2 tell_VV0 us_PPIO2 the_AT direction_NN1 in_II which_DDQ things_NN2 will_VM naturally_RR happen_VVI ._. 
Free_JJ energies_NN2 also_RR provide_VV0 us_PPIO2 with_IW quantitative_JJ information_NN1 about_II processes_NN2 ._. 
The_AT change_NN1 in_II the_AT free_JJ energy_NN1 is_VBZ equal_JJ to_II the_AT maximum_JJ work_NN1 that_CST can_VM be_VBI extracted_VVN from_II a_AT1 spontaneous_JJ process_NN1 ,_, or_CC in_II the_AT case_NN1 of_IO a_AT1 non-spontaneous_JJ processes_NN2 ,_, the_AT minimum_JJ amount_NN1 of_IO work_NN1 that_CST is_VBZ required_VVN to_TO cause_VVI the_AT process_NN1 to_TO occur_VVI ._. 
Figure_NN1 2.1_MC shows_VVZ the_AT pressure-temperature_JJ projection_NN1 of_IO the_AT phase_NN1 diagram_NN1 for_IF a_AT1 general_JJ one-component_NN1 system_NN1 ._. 
Depending_II21 on_II22 the_AT temperature_NN1 and_CC pressure_NN1 ,_, the_AT system_NN1 can_VM exist_VVI in_RP either_RR a_AT1 solid_JJ ,_, liquid_JJ or_CC vapor_NN1 phase_NN1 ._. 
Lines_NN2 separate_VV0 the_AT various_JJ phases_NN2 ._. 
On_II the_AT lines_NN2 ,_, two_MC phases_NN2 coexist_VV0 ._. 
The_AT line_NN1 separating_VVG the_AT vapor_NN1 and_CC liquid_JJ phases_NN2 is_VBZ known_VVN as_II the_AT vapor_NN1 pressure_NN1 curve_NN1 ._. 
On_II crossing_VVG this_DD1 curve_NN1 ,_, the_AT system_NN1 will_VM transform_VVI discontinuously_RR from_II a_AT1 liquid_NN1 to_II a_AT1 vapor_NN1 (_( or_CC vice-versa_RR )_) ._. 
At_II high_JJ temperatures_NN2 ,_, the_AT vapor_NN1 pressure_NN1 curve_NN1 ends_VVZ at_II a_AT1 critical_JJ point_NN1 ._. 
Beyond_II this_DD1 point_NN1 ,_, there_EX is_VBZ no_AT real_JJ distinction_NN1 between_II vapor_NN1 and_CC liquid_JJ phases_NN2 ._. 
By_II going_VVG around_II the_AT critical_JJ point_NN1 ,_, a_AT1 liquid_NN1 can_VM be_VBI continuously_RR transformed_VVN into_II vapor_NN1 ._. 
The_AT line_NN1 separating_VVG the_AT solid_JJ and_CC liquid_JJ phases_NN2 is_VBZ known_VVN as_II the_AT melting_NN1 or_CC freezing_JJ curve_NN1 ._. 
The_AT line_NN1 separating_VVG the_AT solid_JJ and_CC vapor_NN1 phases_NN2 is_VBZ known_VVN as_II the_AT sublimation_NN1 curve_NN1 ._. 
The_AT point_NN1 where_RRQ the_AT vapor_NN1 pressure_NN1 curve_NN1 ,_, the_AT melting_NN1 curve_NN1 ,_, and_CC the_AT sublimation_NN1 curves_NN2 meet_VV0 is_VBZ the_AT triple_JJ point_NN1 ._. 
At_II these_DD2 conditions_NN2 ,_, the_AT solid_JJ ,_, liquid_NN1 ,_, and_CC vapor_NN1 phases_NN2 can_VM simultaneously_RR coexist_VVI ._. 
In_II figure_NN1 2.2_MC ,_, we_PPIS2 show_VV0 the_AT temperature-density_JJ phase_NN1 diagram_NN1 for_IF a_AT1 general_JJ pure_JJ substance_NN1 ._. 
As_CSA with_IW the_AT pressure-temperature_JJ diagram_NN1 ,_, the_AT temperature-density_JJ phase_NN1 diagram_NN1 is_VBZ divided_VVN by_II various_JJ curves_NN2 into_II vapor_NN1 ,_, liquid_NN1 ,_, and_CC solid_JJ phases_NN2 ._. 
Outside_II these_DD2 curves_NN2 ,_, the_AT system_NN1 exists_VVZ as_II a_AT1 single_JJ phase_NN1 ._. 
The_AT dashed-line_NN1 represents_VVZ the_AT triple_JJ point_NN1 ._. 
Anywhere_RL along_II the_AT dash-line_NN1 ,_, the_AT vapor_NN1 ,_, liquid_JJ and_CC solid_JJ phases_NN2 can_VM simultaneously_RR exist_VVI ._. 
Now_RT let_VM21 's_VM22 derive_VVI the_AT mathematical_JJ conditions_NN2 for_IF equilibrium_NN1 between_II two_MC coexisting_JJ phases_NN2 ._. 
We_PPIS2 consider_VV0 an_AT1 isolated_JJ system_NN1 that_CST is_VBZ separated_VVN into_II two_MC phases_NN2 ,_, which_DDQ we_PPIS2 label_VV0 A_ZZ1 and_CC B._NP1 The_AT volume_NN1 occupied_VVN by_II each_DD1 phase_NN1 can_VM change_VVI ,_, in_RR21 addition_RR22 ,_, the_AT both_DB2 phases_NN2 can_VM freely_RR exchange_VVI energy_NN1 and_CC material_NN1 with_IW each_PPX221 other_PPX222 ._. 
Because_CS the_AT system_NN1 is_VBZ isolated_VVN ,_, the_AT total_JJ energy_NN1 U_ZZ1 ,_, the_AT total_JJ volume_NN1 V_ZZ1 ,_, and_CC the_AT total_JJ number_NN1 of_IO moles_NN2 N_ZZ1 in_II the_AT system_NN1 must_VM remain_VVI constant_JJ ._. 
In_II this_DD1 section_NN1 ,_, we_PPIS2 derive_VV0 the_AT Clapeyron_NN1 equation_NN1 ._. 
This_DD1 equation_NN1 relates_VVZ changes_NN2 in_II the_AT pressure_NN1 to_II changes_NN2 in_II the_AT temperature_NN1 along_II a_AT1 two-phase_JJ coexistence_NN1 curve_NN1 ._. 
This_DD1 is_VBZ one_MC1 form_NN1 of_IO the_AT Clapeyron_NN1 equation_NN1 ._. 
It_PPH1 relates_VVZ the_AT slope_NN1 of_IO coexistence_NN1 curve_NN1 to_II the_AT entropy_NN1 change_NN1 and_CC volume_NN1 change_NN1 of_IO the_AT phase_NN1 transition_NN1 ._. 
Entropy_NN1 is_VBZ not_XX directly_RR measureable_JJ ,_, and_CC ,_, therefore_RR ,_, the_AT Clapeyron_NN1 equation_NN1 as_CSA written_VVN above_RL is_VBZ not_XX in_II a_AT1 convenient_JJ form_NN1 ._. 
However_RR ,_, we_PPIS2 can_VM relate_VVI entropy_NN1 changes_NN2 to_II enthalpy_NN1 changes_NN2 ,_, which_DDQ can_VM be_VBI directly_RR measured_VVN ._. 
Thus_RR ,_, the_AT entropy_NN1 change_NN1 of_IO a_AT1 phase_NN1 transition_NN1 ,_, which_DDQ is_VBZ not_XX directly_RR measureable_JJ ,_, can_VM be_VBI determined_VVN from_II the_AT enthalpy_NN1 change_NN1 of_IO the_AT phase_NN1 transition_NN1 ,_, which_DDQ is_VBZ directly_RR measurable_JJ ._. 
The_AT conditions_NN2 for_IF phase_NN1 equilibria_NN2 can_VM also_RR be_VBI extended_VVN to_II multicomponent_JJ systems_NN2 ._. 
How_RGQ many_DA2 variables_NN2 need_VV0 to_TO be_VBI specified_VVN in_BCL21 order_BCL22 to_TO fix_VVI the_AT state_NN1 of_IO a_AT1 system_NN1 ?_? 
In_BCL21 order_BCL22 to_TO fix_VVI the_AT state_NN1 of_IO a_AT1 one-phase_JJ system_NN1 ,_, the_AT composition_NN1 of_IO the_AT phase_NN1 must_VM be_VBI specified_VVN as_II31 well_II32 as_II33 two_MC additional_JJ intensive_JJ variables_NN2 ._. 
The_AT maximum_JJ number_NN1 of_IO degrees_NN2 of_IO freedom_NN1 that_CST a_AT1 system_NN1 can_VM have_VHI is_VBZ given_VVN when_CS there_EX is_VBZ only_RR one_MC1 phase_NN1 present_NN1 ._. 
For_IF a_AT1 binary_JJ mixture_NN1 ,_, we_PPIS2 find_VV0 that_CST there_EX are_VBR at_RR21 most_RR22 three_MC degrees_NN2 of_IO freedom_NN1 ._. 
This_DD1 means_VVZ that_CST we_PPIS2 can_VM represent_VVI the_AT state_NN1 of_IO binary_JJ mixture_NN1 using_VVG a_AT1 three_MC dimensional_JJ diagram_NN1 ._. 
An_AT1 example_NN1 of_IO such_DA a_AT1 diagram_NN1 is_VBZ given_VVN in_II Fig._NN1 3.1_MC ,_, which_DDQ is_VBZ for_IF mixtures_NN2 of_IO methane_NN1 and_CC ethane_NN1 ._. 
The_AT key_JJ feature_NN1 of_IO this_DD1 phase_NN1 diagram_NN1 is_VBZ a_AT1 solid_JJ body_NN1 in_II the_AT center_NN1 of_IO the_AT figure_NN1 ._. 
Within_II this_DD1 solid_JJ body_NN1 ,_, the_AT system_NN1 exists_VVZ as_II a_AT1 two-phase_JJ mixture_NN1 ,_, with_IW a_AT1 coexisting_JJ liquid_NN1 and_CC vapor_NN1 phase_NN1 ._. 
Above_II this_DD1 body_NN1 ,_, the_AT system_NN1 exists_VVZ as_II a_AT1 single_JJ liquid_JJ phase_NN1 ;_; below_II this_DD1 body_NN1 ,_, the_AT system_NN1 is_VBZ a_AT1 single_JJ vapor_NN1 phase_NN1 ._. 
The_AT upper_JJ surface_NN1 that_CST bounds_VVZ the_AT body_NN1 is_VBZ the_AT locus_NN1 of_IO bubble_NN1 points_NN2 (_( i.e._REX ,_, the_AT points_NN2 at_II which_DDQ bubbles_NN2 begin_VV0 to_TO appear_VVI in_II a_AT1 liquid_NN1 )_) ._. 
The_AT lower_JJR surface_NN1 (_( marked_VVN by_II green_JJ points_NN2 )_) is_VBZ the_AT locus_NN1 of_IO dew_NN1 points_NN2 (_( i.e._REX ,_, the_AT points_NN2 at_II which_DDQ droplets_NN2 begin_VV0 to_TO appear_VVI in_II a_AT1 vapor_NN1 )_) ._. 
The_AT points_NN2 C1_FO and_CC C2_FO are_VBR the_AT critical_JJ points_NN2 of_IO pure_JJ methane_NN1 and_CC ethane_NN1 ,_, respectively_RR ._. 
The_AT line_NN1 connecting_VVG these_DD2 two_MC points_NN2 ,_, which_DDQ is_VBZ the_AT intersection_NN1 of_IO the_AT bubble_NN1 point_NN1 and_CC dew_NN1 point_NN1 surface_NN1 ,_, is_VBZ the_AT critical_JJ locus_NN1 ._. 
This_DD1 is_VBZ the_AT set_NN1 of_IO critical_JJ points_NN2 for_IF the_AT various_JJ mixtures_NN2 of_IO methane_NN1 and_CC ethane_NN1 ._. 
The_AT black_JJ curve_NN1 connecting_VVG points_NN2 A_ZZ1 and_CC C1_FO is_VBZ the_AT vapor_NN1 pressure_NN1 curve_NN1 of_IO pure_JJ methane_NN1 ,_, and_CC the_AT violet_JJ curve_NN1 connecting_VVG points_NN2 B_ZZ1 and_CC is_VBZ the_AT vapor_NN1 pressure_NN1 curve_NN1 of_IO pure_JJ ethane_NN1 ._. 
For_IF a_AT1 one_MC1 component_NN1 system_NN1 ,_, the_AT bubble_NN1 point_NN1 and_CC the_AT dew_NN1 point_NN1 are_VBR the_AT same_DA and_CC lie_VV0 along_RP the_AT vapor_NN1 pressure_NN1 curve_NN1 ,_, however_RR ,_, this_DD1 is_VBZ not_XX necessarily_RR the_AT case_NN1 for_IF a_AT1 mixture_NN1 ._. 
Within_II envelopes_NN2 contained_VVN between_II the_AT vapor_NN1 pressure_NN1 curves_NN2 of_IO the_AT pure_JJ components_NN2 ,_, a_AT1 mixture_NN1 consists_VVZ of_IO a_AT1 coexisting_JJ vapor_NN1 and_CC liquid_JJ phases_NN2 ._. 
The_AT upper_JJ part_NN1 of_IO the_AT envelope_NN1 (_( the_AT solid_JJ curve_NN1 with_IW filled_JJ symbols_NN2 )_) is_VBZ the_AT bubble_NN1 point_NN1 curve_NN1 ;_; the_AT lower_JJR part_NN1 of_IO the_AT envelope_NN1 (_( the_AT dashed_JJ curve_NN1 with_IW open_JJ symbols_NN2 )_) is_VBZ the_AT dew_NN1 point_NN1 curve_NN1 ._. 
Different_JJ envelopes_NN2 correspond_VV0 to_II different_JJ mixture_NN1 compositions_NN2 ._. 
In_II the_AT systems_NN2 that_CST we_PPIS2 have_VH0 examined_VVN so_RG far_RR ,_, the_AT bubble_NN1 point_NN1 and_CC the_AT dew_NN1 point_NN1 of_IO the_AT mixture_NN1 vary_VV0 monotonically_RR with_IW the_AT composition_NN1 ._. 
This_DD1 is_VBZ the_AT case_NN1 for_IF ideal_JJ systems_NN2 ._. 
However_RR ,_, for_IF very_RG non-ideal_JJ systems_NN2 ,_, there_EX may_VM be_VBI a_AT1 maximum_NN1 or_CC a_AT1 minimum_NN1 in_II the_AT bubble_NN1 and_CC dew_NN1 point_NN1 curves_NN2 ._. 
This_DD1 is_VBZ the_AT case_NN1 for_IF azeotropic_JJ systems_NN2 ._. 
An_AT1 example_NN1 of_IO a_AT1 system_NN1 that_CST exhibits_VVZ a_AT1 low-boiling_JJ azeotrope_NN1 is_VBZ a_AT1 mixture_NN1 of_IO n-heptane_NN1 and_CC ethanol_NN1 ,_, which_DDQ is_VBZ shown_VVN in_II Figure_NN1 3.5_MC ._. 
For_IF this_DD1 type_NN1 of_IO system_NN1 ,_, both_DB2 the_AT bubble_NN1 and_CC the_AT dew_NN1 point_NN1 temperature_NN1 curves_NN2 have_VH0 a_AT1 local_JJ minimum_NN1 at_II the_AT same_DA composition_NN1 ._. 
At_II this_DD1 composition_NN1 ,_, these_DD2 two_MC curves_NN2 meet_VV0 ._. 
This_DD1 point_NN1 is_VBZ known_VVN as_II the_AT azeotrope_NN1 ._. 
At_II the_AT azeotrope_NN1 ,_, the_AT composition_NN1 of_IO the_AT coexisting_JJ liquid_NN1 and_CC vapor_NN1 phases_NN2 are_VBR identical_JJ ._. 
In_II this_DD1 case_NN1 at_II the_AT azeotrope_NN1 ,_, the_AT boiling_JJ temperature_NN1 of_IO the_AT liquid_NN1 is_VBZ lower_JJR than_CSN the_AT boiling_JJ temperature_NN1 of_IO either_DD1 pure_JJ components_NN2 ._. 
The_AT corresponding_JJ bubble_NN1 and_CC dew_NN1 point_NN1 pressure_NN1 curves_NN2 have_VH0 a_AT1 maximum_NN1 at_II the_AT azeotrope_NN1 ._. 
When_CS two_MC liquids_NN2 are_VBR mixed_VVN together_RL ,_, they_PPHS2 do_VD0 not_XX always_RR form_VVI a_AT1 single_JJ ,_, homogenous_JJ liquid_JJ phase_NN1 ._. 
In_II many_DA2 cases_NN2 ,_, two_MC liquid_JJ phases_NN2 are_VBR formed_VVN ,_, with_IW one_MC1 phase_NN1 richer_JJR in_II the_AT first_MD component_NN1 and_CC the_AT other_JJ phase_NN1 richer_JJR in_II the_AT second_MD component_NN1 ._. 
The_AT classic_JJ example_NN1 of_IO a_AT1 system_NN1 that_CST exhibits_VVZ this_DD1 behavior_NN1 is_VBZ a_AT1 mixture_NN1 of_IO oil_NN1 and_CC water_NN1 ._. 
The_AT maximum_NN1 of_IO the_AT liquid-liquid_JJ phase_NN1 envelope_NN1 and_CC is_VBZ known_VVN as_II the_AT critical_JJ point_NN1 of_IO the_AT mixture_NN1 ._. 
Above_II the_AT critical_JJ temperature_NN1 (_( i.e._REX ,_, the_AT temperature_NN1 at_II the_AT critical_JJ point_NN1 )_) ,_, the_AT system_NN1 exists_VVZ as_II a_AT1 single_JJ liquid_JJ phase_NN1 ._. 
Below_II the_AT critical_JJ temperature_NN1 ,_, the_AT system_NN1 can_VM split_VVI into_II two_MC coexisting_VVG liquid_JJ phases_NN2 ,_, depending_II21 on_II22 the_AT overall_JJ composition_NN1 ._. 
The_AT basic_JJ reason_NN1 why_RRQ liquid-liquid_JJ phase_NN1 separation_NN1 occurs_VVZ is_VBZ that_CST the_AT attractive_JJ interactions_NN2 between_II different_JJ molecules_NN2 are_VBR weaker_JJR than_CSN the_AT attractive_JJ interactions_NN2 between_II similar_JJ molecules_NN2 ._. 
As_II a_AT1 result_NN1 ,_, similar_JJ molecules_NN2 prefer_VV0 to_TO be_VBI near_II21 to_II22 each_PPX221 other_PPX222 and_CC than_CSN to_II dissimilar_JJ molecules_NN2 ._. 
As_II the_AT pressure_NN1 of_IO the_AT system_NN1 decreases_VVZ ,_, the_AT boiling_JJ temperature_NN1 decreases_VVZ ,_, in_RR21 general_RR22 ._. 
Therefore_RR ,_, we_PPIS2 expect_VV0 the_AT vapor-liquid_JJ coexistence_NN1 envelope_NN1 to_TO drop_VVI to_TO lower_VVI temperatures_NN2 as_II the_AT pressures_NN2 decrease_VV0 ._. 
Changes_NN2 in_II pressure_NN1 ,_, however_RR ,_, do_VD0 not_XX have_VHI a_AT1 strong_JJ influence_NN1 on_II the_AT phase_NN1 behavior_NN1 of_IO liquids_NN2 ._. 
As_II a_AT1 result_NN1 ,_, we_PPIS2 do_VD0 not_XX expect_VVI the_AT liquid-liquid_JJ phase_NN1 envelope_NN1 to_TO change_VVI much_RR with_IW pressure_NN1 ._. 
In_II many_DA2 situations_NN2 ,_, we_PPIS2 need_VV0 to_TO predict_VVI the_AT properties_NN2 of_IO a_AT1 mixture_NN1 ,_, given_CS21 that_CS22 we_PPIS2 already_RR know_VV0 the_AT properties_NN2 of_IO the_AT pure_JJ species_NN ._. 
To_TO do_VDI this_DD1 requires_VVZ a_AT1 model_NN1 that_CST can_VM describe_VVI how_RGQ various_JJ components_NN2 mix_VV0 ._. 
In_II mathematical_JJ terms_NN2 ,_, this_DD1 means_VVZ that_CST we_PPIS2 need_VV0 to_TO relate_VVI the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO a_AT1 mixture_NN1 to_II the_AT Gibbs_NP1 free_JJ energy_NN1 of_IO the_AT various_JJ pure_JJ components_NN2 ._. 
One_MC1 of_IO the_AT simplest_JJT models_NN2 that_CST achieves_VVZ this_DD1 is_VBZ the_AT ideal_JJ solution_NN1 mode_NN1 ._. 
In_II this_DD1 lecture_NN1 ,_, we_PPIS2 present_VV0 the_AT ideal_JJ solution_NN1 model_NN1 ._. 
Then_RT we_PPIS2 apply_VV0 this_DD1 model_NN1 to_TO describe_VVI vapor-liquid_JJ equilibria_NN2 ,_, and_CC as_II a_AT1 result_NN1 ,_, derive_VV0 Raoult_NP1 's_GE law_NN1 ._. 
In_II a_AT1 ideal_JJ solution_NN1 ,_, we_PPIS2 see_VV0 that_CST the_AT chemical_JJ potential_NN1 of_IO a_AT1 species_NN depends_VVZ on_II its_APPGE mole_NN1 fraction_NN1 and_CC not_XX directly_RR on_II the_AT composition_NN1 of_IO the_AT other_JJ components_NN2 in_II the_AT system_NN1 ._. 
Also_RR ,_, we_PPIS2 see_VV0 that_DD1 mixing_NN1 causes_VVZ the_AT chemical_JJ potential_NN1 of_IO each_DD1 component_NN1 to_TO decrease_VVI ._. 
