SUM02.pdf - 2 Thermodynamics Summary Extensive and...

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2 Thermodynamics : Summary Extensive and intensive variables : The equilibrium state of a thermodynamic system is char- acterized by specifying a number of state variables which can be either extensive (scaling lin- early with system size), or intensive (scaling as the zeroth power of system size). Extensive quantities include: energy E , entropy S , particle number N , magnetization M , etc. Inten- sive quantities include temperature T , pressure p , number density n , magnetic field H , etc. The ratio of two extensive quantities is intensive, e.g. n = N/V . In the thermodynamic limit , all extensive state variables tend to infinity (in whatever units are appropriate), while their various ratios are all finite. A full description of the state of any thermodynamic sys- tem must involve at least one extensive variable (but may or may not include intensive variables). Work : The internal energy of a thermodynamic system can change as a result of a gener- alized displacement dX i , as a result of work W done by the system. We write the differential form of W as ¯ dW = summationdisplay i y i dX i summationdisplay a µ a dN a , where y i is the generalized force conjugate to the generalized displacement X i , and µ a is the chemical potential of species a , which is conjugate to the number of particles of that species, N a . Think of chemical work as the work required to assemble particles out of infinitely remote constituents. The slash through the differential symbol indicates that ¯ dW is an inexact differential , i.e. there is no function W ( T,p,V,... ) . Heat : Aside from work done by or on the system, there is another way of changing the system’s internal energy, which is by transferring heat , Q . Heat is a form of energy contained in the random microscopic motions of the constituent particles. Like ¯ dW , the differential ¯ dQ is also inexact, and there is no heat function Q ( T,p,V,... ) . Transfer of heat under conditions of constant volume or pressure and constant particle number results in a change of the the thermodynamic state via a change in temperature: dT = ¯ dQ/C , where C is the heat capacity of the system at fixed volume/pressure and particle number. First Law : The First Law of Thermodynamics is a statement of energy conservation which accounts for both types of energies: Δ E = Q W , or in differential form dE = ¯ dQ ¯ dW . Single component systems : A single component system is completely specified by three state variables, which can be taken to be E , V , and N , and writing ¯ dW = pdV µdN , we have ¯ dQ = dE + pdV µdN. If, for example, we want to use variables ( T,V,N ) , we write dE = parenleftbigg ∂E ∂T parenrightbigg V,N dT + parenleftbigg ∂E ∂V parenrightbigg T,N dV + parenleftbigg ∂E ∂N parenrightbigg T,V dN.
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