Standard molar entropy

<h2 id="thermodynamics">Thermodynamics</h2>
<p>If a <a href="/facts/Mole_(unit)/kCMFwCc0">mole</a> of a solid substance is a perfectly ordered solid at 0 K, then if the solid is warmed by its surroundings to 298.15 K without melting, its absolute molar entropy would be the sum of a series of N stepwise and reversible entropy changes. The limit of this sum as 
  
    
      
        N
        →
        ∞
      
    
    {\displaystyle N\rightarrow \infty }
  
 becomes an integral:
</p>

S
          
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            N
          
        
        Δ
        
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              d
              
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                2
              
            
          
        
        
          
            
              d
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        d
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            T
          
        
        d
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    {\displaystyle S^{\circ }=\sum _{k=1}^{N}\Delta S_{k}=\sum _{k=1}^{N}{\frac {dQ_{k}}{T}}\rightarrow \int _{0}^{T_{2}}{\frac {dS}{dT}}dT=\int _{0}^{T_{2}}{\frac {C_{p_{k}}}{T}}dT}

<p>In this example, 
  
    
      
        
          T
          
            2
          
        
        =
        298.15
        K
      
    
    {\displaystyle T_{2}=298.15K}
  
 and 
  
    
      
        
          C
          
            
              p
              
                k
              
            
          
        
      
    
    {\displaystyle C_{p_{k}}}
  
 is the <a href="/facts/Molar_heat_capacity/o18zBgWt">molar heat capacity</a> at a constant pressure of the substance in the <a href="/facts/Reversible_process_(thermodynamics)/RtAnDT3C">reversible process</a> k. The molar heat capacity is not constant during the experiment because it changes depending on the (increasing) temperature of the substance. Therefore, a table of values for 
  
    
      
        
          
            
              C
              
                
                  p
                  
                    k
                  
                
              
            
            T
          
        
      
    
    {\displaystyle {\frac {C_{p_{k}}}{T}}}
  
 is required to find the total molar entropy. The quantity

d
              
                Q
                
                  k
                
              
            
            T
          
        
      
    
    {\displaystyle {\frac {dQ_{k}}{T}}}
  
 represents the ratio of a very small exchange of heat energy to the temperature T. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process.
</p>
<h2 id="chemistry">Chemistry</h2>
<p>The standard molar entropy of a gas at <a href="/facts/Standard_temperature_and_pressure/JhJv9KFJ">STP</a> includes contributions from:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<ul><li>The <a href="/facts/Heat_capacity/I63nzSyp">heat capacity</a> of one mole of the solid from 0 K to the <a href="/facts/Melting_point/MY0w0h0k">melting point</a> (including heat absorbed in any changes between different <a href="/facts/Crystal_structure/bJ4bCeHd">crystal structures</a>).</li>
<li>The <a href="/facts/Latent_heat_of_fusion/eLJIURcp">latent heat of fusion</a> of the solid.</li>
<li>The heat capacity of the liquid from the melting point to the <a href="/facts/Boiling_point/E2BK9QoS">boiling point</a>.</li>
<li>The <a href="/facts/Latent_heat_of_vaporization/TbEWQDNl">latent heat of vaporization</a> of the liquid.</li>
<li>The heat capacity of the gas from the boiling point to room temperature.</li></ul>
<p>Changes in entropy are associated with <a href="/facts/Phase_transitions/eLP2BY8R">phase transitions</a> and <a href="/facts/Chemical_reactions/GpqWlKrz">chemical reactions</a>. <a href="/facts/Chemical_equations/S6QsBnm9">Chemical equations</a> make use of the standard molar entropy of <a href="/facts/Reactants/c3GLInCu">reactants</a> and <a href="/facts/Product_(chemistry)/8uCoDUdv">products</a> to find the standard entropy of reaction:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

Δ
            
              S
              
                ∘
              
            
          
          
            r
            x
            n
          
        
        =
        
          S
          
            p
            r
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            d
            u
            c
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            s
          
          
            ∘
          
        
        −
        
          S
          
            r
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            t
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            ∘
          
        
      
    
    {\displaystyle {\Delta S^{\circ }}_{rxn}=S_{products}^{\circ }-S_{reactants}^{\circ }}

<p>The standard entropy of reaction helps determine whether the reaction will take place <a href="/facts/Spontaneous_process/xyGhdUG0">spontaneously</a>. According to the <a href="/facts/Second_law_of_thermodynamics/m7SJ2eeH">second law of thermodynamics</a>, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings:
</p>

(
        Δ
        
          S
          
            t
            o
            t
            a
            l
          
        
        =
        Δ
        
          S
          
            s
            y
            s
            t
            e
            m
          
        
        +
        Δ
        
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            u
            r
            r
            o
            u
            n
            d
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            n
            g
            s
          
        
        )
        >
        0
      
    
    {\displaystyle (\Delta S_{total}=\Delta S_{system}+\Delta S_{surroundings})>0}

<p>Molar entropy is not the same for all gases. Under identical conditions, it is greater for a heavier gas.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Entropy/s52IXeFz">Entropy</a></li>
<li><a href="/facts/Heat/FjDGssCs">Heat</a></li>
<li><a href="/facts/Gibbs_free_energy/bvenDuzh">Gibbs free energy</a></li>
<li><a href="/facts/Helmholtz_free_energy/2AK0HpPe">Helmholtz free energy</a></li>
<li><a href="/facts/Standard_state/adrN3C0a">Standard state</a></li>
<li><a href="/facts/Third_law_of_thermodynamics/25703lR0">Third law of thermodynamics</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://chem.libretexts.org/Bookshelves/General_Chemistry/Chemistry_1e_(OpenSTAX)/Appendices/Standard_Thermodynamic_Properties_for_Selected_Substances">Table of Standard Thermodynamic Properties for Selected Substances</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Pauling, Linus (1960). The Nature of the Chemical Bond (3rd ed.). Ithaca, NY: Cornell University Press. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kosanke, K. (2004). "Chemical Thermodynamics". Pyrotechnic chemistry. Journal of Pyrotechnics. p. 29. ISBN 1-889526-15-0. <a href="1-889526-15-0" target="_blank">1-889526-15-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Chang, Raymond; Cruickshank, Brandon (2005). "Entropy, Free Energy and Equilibrium". Chemistry. McGraw-Hill Higher Education. p. 765. ISBN 0-07-251264-4. <a href="0-07-251264-4" target="_blank">0-07-251264-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Standard molar entropy open-in-new

Standard molar entropy