Thermodynamic Calculus Manipulations Name Applies to

ChE 210A Thermodynamics and statistical mechanics Extensive thermodynamic potentials (single component) name independent variables differential form integrated form entropy ͍ʚ̿, ͐, ͈ʛ 1 ͊ ̿ ͊͐ ͈ ͍͘ Ɣ ̿͘ ƍ ͐͘ Ǝ ͈͘ ͍ Ɣ ƍ Ǝ ͎ ͎ ͎ ͎ ͎ ͎ energy ̿ʚ͍, ͐, ͈ʛ ̿͘ Ɣ ͎͍͘ Ǝ ͊͐͘ ƍ ͈͘ ̿ Ɣ ͎͍ Ǝ ͊͐ ƍ ͈ enthalpy ͂ʚ͍, ͊, ͈ʛ ͂͘ Ɣ ͎͍͘ ƍ ͐͊͘ ƍ ͈͘ ͂ Ɣ ̿ ƍ ͊͐ Ɣ ͎͍ ƍ ͈ Helmholtz free energy ̻ʚ͎, ͐, ͈ʛ ̻͘ Ɣ Ǝ͍͎͘ Ǝ ͊͐͘ ƍ ͈͘ ̻ Ɣ ̿ Ǝ ͎͍ Ɣ Ǝ͊͐ ƍ ͈ Gibbs free energy ́ʚ͎, ͊, ͈ʛ ́͘ Ɣ Ǝ͍͎͘ ƍ ͐͊͘ ƍ ͈͘ ́ Ɣ ̿ ƍ ͊͐ Ǝ ͎͍ Ɣ ̻ ƍ ͊͐ Ɣ ͂ Ǝ ͎͍ Ɣ ͈ Intensive thermodynamic potentials (single component) name independent variables differential form integrated relations entropy per -particle ͧʚ͙, ͪʛ 1 ͊ ͙ ͊ͪ ͧ͘ Ɣ ͙͘ ƍ ͪ͘ Ɣ Ǝͧ ƍ ƍ ͎ ͎ ͎ ͎ ͎ energy per -particle ͙ʚͧ, ͪʛ ͙͘ Ɣ ͎ͧ͘ Ǝ ͊ͪ͘ Ɣ ͙ Ǝ ͎ͧ ƍ ͊ͪ enthalpy per -particle ͜ʚͧ, ͊ʛ ͘͜ Ɣ ͎ͧ͘ ƍ ͪ͊͘ ͜ Ɣ ͙ ƍ ͊ͪ Ɣ ͜ Ǝ ͎ͧ Helmholtz free energy per -particle ͕ʚ͎, ͪʛ ͕͘ Ɣ Ǝ͎ͧ͘ Ǝ ͊ͪ͘ ͕ Ɣ ͙ Ǝ ͎ͧ Ɣ ͕ ƍ ͊ͪ Gibbs free energy per -particle ͛ʚ͎, ͊ʛ ͛͘ Ɣ Ǝ͎ͧ͘ ƍ ͪ͊͘ ͛ Ɣ ͙ ƍ ͊ͪ Ǝ ͎ͧ Ɣ ͕ ƍ ͊ͪ Ɣ ͜ Ǝ ͎ͧ Ɣ ͛ ChE 210A Thermodynamics and statistical mechanics Measurable quantities name definition name definition pressure temperature ͊ ͎ volume heat of phase change constant -volume heat capacity ͐ isothermal compressibility Δ͂ ̿͘ 1 ͐͘ ͘ ln ͐ ̽ Ɣ ƴ Ƹ Ɣ Ǝ ƴ Ƹ Ɣ Ǝ ƴ Ƹ constant -pressure heat capacity ͎͘ thermal expansivity ͐ ͊͘ ͊͘ ͂͘ 1 ͐͘ ͘ ln ͐ ̽ Ɣ ƴ Ƹ Ɣ ƴ Ƹ Ɣ ƴ Ƹ ͎͘ ͐ ͎͘ ͎͘ Thermodynamic calculus manipulations name applies to ... functional form example inversion anything ͒͘ ͓͘ ͊͘ ͍͘ ƴ Ƹ Ɣ 1Ɵƴ Ƹ ƴ Ƹ Ɣ 1Ɵƴ Ƹ ͓͘ ͒͘ ͍͘ ͊͘ triple product rule anything ͒͘ ͔͘ ͓͘ ͊͘ ͍͘ ͍͘ ƴ Ƹ ƴ Ƹ ƴ Ƹ Ɣ Ǝ1 ƴ Ƹ Ɣ Ǝ ƴ Ƹ Ơƴ Ƹ ͓͘ ͒͘ ͔͘ ͎͘ ͎͘ ͊͘ Maxwell’s relations potential second ͦ ͦ ͘ ̀ ͘ ̀ ̻͘ ̼͘ ͍͘ ͐͘ derivatives ʦ ʧ Ɣ ʦ ʧ Ŵ ƴ Ƹ Ɣ ƴ Ƹ ƴ Ƹ Ɣ Ǝ ƴ Ƹ ͓͒͘͘ ͓͒͘͘ ͒͘ ͓͘ ͊͘ ͎͘ addition of variable anything ͒͘ ͒͘ ͓͘ ͂͘ ͂͘ ͍͘ ƴ Ƹ Ɣ ƴ Ƹ Ơƴ Ƹ ƴ Ƹ Ɣ ƴ Ƹ Ơƴ Ƹ Ɣ ͎ ͓͘ ͑͘ ͑͘ ͍͘ ͎͘ ͎͘ potential potentials ͘ʚ̀ͥ⁄͒ʛ ̀ͦ ͘ʚ̻⁄͎ʛ ̿ transformation Ɣ Ǝ Ɣ Ǝ ͒͘ ͒ͦ ͎͘ ͎ͦ nonnatural anything ̻͘ ̻͘ ͒͘ ̻͘ ̿͘ ̿͘ ͍͘ ̿͘ ͍͘ derivative ̻ʚ͒, ͓ʛ Ŵ ƴ Ƹ Ɣ ƴ Ƹ ƴ Ƹ ƍ ƴ Ƹ ƴ Ƹ Ɣ ƴ Ƹ ƴ Ƹ ƍ ƴ Ƹ Ɣ ͎ ƴ Ƹ Ǝ ͊ ͓͘ ͒͘ ͓͘ ͓͘ ͐͘ ͍͘ ͐͘ ͐͘ ͐͘ Note: Sometimes the notation is used instead of to indicate the thermodynamic internal energy. ͏ ̿ ChE 210A Thermodynamics and statistical mechanics Derivations and memory techniques Triple product rule Take, as an example, the function , although we could consider any state function. We know that the full differential is given by͊ʚ͎, ͐ʛ ͊͘ ͊͘ ͊͘ Ɣ ƴ Ƹ ͎͘ ƍ ƴ Ƹ ͐͘ ͎͘ ͐͘ Now consider the case in which is held constant, such that . Then we can combine the differentials and at constant͊ conditions. ͊͘ Ɣ 0 ͎͘ ͐͘ ͊ ͊͘ ͎͘ ͊͘ 0 Ɣ ƴ Ƹ ƴ Ƹ ƍ ƴ Ƹ ͎͘ ͐͘ ͐͘ Rearranging and using the inversion rule gives the triple product rule: ͊͘ ͐͘ ͎͘ ƴ Ƹ ƴ Ƹ ƴ Ƹ Ɣ Ǝ1 ͎͘ ͊͘ ͐͘ How can you remember this? It’s simple. Starting with one partial derivative, just rotate the variables by putting the constant one in the numerator, the numerator in the denominator, and the denominator in the constant. Do two rotations and remember the product is -1. Maxwell’s relations For constant conditions, as we have discussed here, there is a particularly convenient way to remember all͈ the Maxwell relationships without having to go through all the derivations each time (although you should always be able to do so!). It’s called the “magic square”. Here is what it looks like: G P H T S A V U To get Maxwell relations from the magic square, you need to do draw arrows that swoop past the thermodynamic potentials along the side edges (not corners). Here is an example: G P H T S A V U The arrows indicate that . They both start at the numerator and end at the constant ʠ ʡ Ɣ Ǝ ʠ ʡ value. The potential they swoop by is , which is also the potential from which this Maxwell relation is ́ ChE 210A Thermodynamics and statistical mechanics derived. The negative sign comes from the fact that is in the top-left quadrant (think of it as negative x and positive y). Maxwell relations about alsó have a negative sign, but those for and do not. ͏ ʚ̿ʛ ͂ ̻ To remember the order of the magic square, start at the top left hand corner and use the mnemonic “Great people have studied under very able teachers.” (No egoism intended.) Potential transformation As an example, let’s prove ʚ⁄ ʛ . To do this, start with the fact that ƔƎ v ̿ Ɣ ̻ƍ͎͍ Now, express as the derivative of : ͍ ̻ ̻͘ ̿ Ɣ ̻ Ǝ ͎ ƴ Ƹ ͎͘ Divide both sides by ͦ ͎ : ̿ ̻ 1 ̻͘ ͦ Ɣ ͦ Ǝ ƴ Ƹ ͎ ͎ ͎ ͎͘ Now, the RHS can be expressed as a derivative using the chain rule, giving finally: ̿ ͘ʚ̻⁄ ͎ ʛ ƔƎ ʦ ʧ ͎ͦ ͎͘ Nonnatural derivative What if we know , but want to find ? Here, is not one of the natural variables that can ̿ʚ͍, ͐ʛ ʠ ʡ ͊ be held constant. We start by writing the full differential : ̿͘ ̿͘ ̿͘ ̿͘ Ɣ ƴ Ƹ ͍͘ ƍ ƴ Ƹ ͐͘ ͍͘ ͐͘ Then, we divide through by the desired derivative, taken at constant : ͊ ̿͘ ̿͘ ͍͘ ̿͘ ƴ Ƹ Ɣ ƴ Ƹ ƴ Ƹ ƍ ƴ Ƹ ͐͘ ͍͘ ͐͘ ͐͘ Substituting in some definitions gives, ̿͘ ͍͘ ƴ Ƹ Ɣ ͎ ƴ Ƹ Ǝ ͊ ͐͘ ͐͘ ChE 210A Thermodynamics and statistical mechanics This procedure works with any complete state function, even with nonnatural variables. For example, starting with E , we could show: ʚ͎, ͐ʛ ̿͘ ̿͘ ͎͘ ̿͘ ƴ Ƹ Ɣ ƴ Ƹ ƴ Ƹ ƍ ƴ Ƹ ͐͘ ͎͘ ͐͘ ͐͘ Carrying this a little further, to get to measurable variables: ̿͘ ̽ ͍͘ ƴ Ƹ Ɣ ƍ ͎ ƴ Ƹ Ǝ ͊ ͐͘ ͐ ͐͘ ̽ ͊͘ Ɣ Ǝ ͎ ƴ Ƹ Ǝ ͊ ͐ ͎͘ ̽ ͊͘ ͐͘ Ɣ ƍ ͎ ƴ Ƹ ƴ Ƹ Ǝ ͊ ͐ ͐͘ ͎͘ ̽ ͎ Ɣ ƍ Ǝ ͊ ͐ Here, another nonnatural derivative was expanded in the first line, a Maxwell relation was used in the second, and the triple product rule was used in the third. .

Load more