Entropy (astrophysics) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-04

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Entropy_(astrophysics)


00:02:29 constant



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SUMMARY
=======
In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows.
Using the first law of thermodynamics for a quasi-static, infinitesimal process for a hydrostatic system

  
    
      
        d
        Q
        =
        d
        U
        −
        d
        W
        .
        
      
    
    {\displaystyle dQ=dU-dW.\,}
  For an ideal gas in this special case, the internal energy, U, is only a function of the temperature T; therefore the partial derivative of heat capacity with respect to T is identically the same as the full derivative, yielding through some manipulation

  
    
      
        d
        Q
        =
        
          C
          
            V
          
        
        d
        T
        +
        P
        
        d
        V
        .
      
    
    {\displaystyle dQ=C_{V}dT+P\,dV.}
  Further manipulation using the differential version of the ideal gas law, the previous equation, and assuming constant pressure, one finds

  
    
      
        d
        Q
        =
        
          C
          
            P
          
        
        d
        T
        −
        V
        
        d
        P
        .
      
    
    {\displaystyle dQ=C_{P}dT-V\,dP.}
  For an adiabatic process 
  
    
      
        d
        Q
        =
        0
        
      
    
    {\displaystyle dQ=0\,}
   and recalling 
  
    
      
        γ
        =
        
          
            
              C
              
                P
              
            
            
              C
              
                V
              
            
          
        
        
      
    
    {\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}\,}
  , one finds

One can solve this simple differential equation to find

  
    
      
        P
        
          V
          
            γ
          
        
        =
        
          constant
        
        =
        K
        
      
    
    {\displaystyle PV^{\gamma }={\text{constant}}=K\,}
  This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows

  
    
      
        P
        =
        
          
            
              ρ
              
                k
                
                  B
                
              
              T
            
            
              μ
              
                m
                
                  H
                
              
            
          
        
        ,
      
    
    {\displaystyle P={\frac {\rho k_{B}T}{\mu m_{H}}},}
  where 
  
    
      
        
          k
          
            B
          
        
        
      
    
    {\displaystyle k_{B}\,}
   is Boltzmann's constant.
Substituting this into the above equation along with 
  
    
      
        V
        =
        [
        g
        r
        a
        m
        s
        ]
        
          /
        
        ρ
        
      
    
    {\displaystyle V=[grams]/\rho \,}
   and 
  
    
      
        γ
        =
        5
        
          /
        
        3
        
      
    
    {\displaystyle \gamma =5/3\,}
   for an ideal monatomic gas one finds

  
    
      
        K
        =
        
          
            
              
                k
                
                  B
                
              
              T
            
            
              (
              ρ
              
                /
              
              μ
              
                m
                
                  H
                
              
              
                )
               ...