Fix comment

09ba625b · Ian Bell · b3fd35eb · 09ba625b
Commit 09ba625b authored 2 years ago by Ian Bell
--- a/doc/source/derivs/derivs.ipynb
+++ b/doc/source/derivs/derivs.ipynb
@@ -269,7 +269,7 @@
    }
   ],
   "source": [
-    "model.get_dmBnvirdTm(2, 3, 300, z) # third temperature derivative of the "
+    "model.get_dmBnvirdTm(2, 3, 300, z) # third temperature derivative of the second virial coefficient"
   ]
  }
 ],

 %% Cell type:markdown id:1afc4517 tags:
 # Thermodynamic Derivatives
 ## Helmholtz energy derivatives
 Thermodynamic derivatives are at the very heart of teqp.  All models are defined in the form $\alpha^r(T, \rho, z)$, where $\rho$ is the molar density, and z are mole fractions.  There are exceptions for models for which the independent variables are in simulation units (Lennard-Jones and its ilk).
 Thereofore, to obtain the residual pressure, it is obtained as a derivative:
 $$ p^r = \rho R T \left( \rho \left(\frac{\partial \alpha^r}{\partial \rho}\right)_{T}\right)$$
 and other residual thermodynamic properties are defined likewise.
 We can define the concise derivative
 $$ \Lambda^r_{xy} = (1/T)^x(\rho)^y\left(\frac{\partial^{x+y}(\alpha^r)}{\partial (1/T)^x\partial \rho^y}\right)$$
 so we can re-write the derivative above as
 $$ p^r = \rho R T \Lambda^r_{01}$$
 Similar definitions apply for all the other residual thermodynamic properties
 %% Cell type:code id:ca35cc29 tags:
 ``` python
 import teqp
 import numpy as np
 ```
 %% Cell type:code id:8d254c40 tags:
 ``` python
 Tc_K = [300]
 pc_Pa = [4e6]
 acentric = [0.01]
 model = teqp.canonical_PR(Tc_K, pc_Pa, acentric)
 ```
 %% Cell type:code id:49ee3453 tags:
 ``` python
 z = np.array([1.0])
 model.get_Ar01(300,300,z)
 ```
 %% Output
    -0.06836660379313926
 %% Cell type:markdown id:498a2350 tags:
 And there are additional methods to obtain all the derivatives up to a given order:
 %% Cell type:code id:3ae755e1 tags:
 ``` python
 model.get_Ar06n(300,300,z) # derivatives 00, 01, 02, ... 06
 ```
 %% Output
    [-0.06966138343515413,
     -0.06836660379313926,
     0.0025357822532378147,
     -0.00015701162203571184,
     1.6818628788290574e-05,
     -2.2305940927885907e-06,
     3.8259258513417917e-07]
 %% Cell type:markdown id:cd644a3e tags:
 But more derivatives are slower than fewer:
 %% Cell type:code id:4e0609ae tags:
 ``` python
 %timeit model.get_Ar01(300,300,z)
 %timeit model.get_Ar04n(300,300,z)
 ```
 %% Output
    1.9 µs ± 38.5 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
    2.07 µs ± 24.8 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
 %% Cell type:markdown id:d05e9070 tags:
 Note: calling overhead is usually on the order of 1 microsecond
 %% Cell type:markdown id:502e9830 tags:
 ## Virial coefficients
 Virial coefficients represent the thermodynamics of the interaction of two-, three-, ... bodies interacting with each other.  They can be obtained rigorously if the potential energy surface of interaction is fully known.  In general, such a surface can only be constructed for small rigid molecules.  Many simple thermodynamic models do a poor job of predicting the thermodynamics captured by the virial coefficients.
 The i-th virial coefficient is defined by
 $$
 	B_i = \frac{(\alpha^r)^{(i-1)}}{(i-2)!}
 $$
 with the concise derivative term
 $$
 	(\alpha^r)^{(i)} = \lim_{\rho \to 0} \left(\frac{\partial^{i}\alpha^r}{\partial \rho^{i}}\right)_{T,\vec{x}}
 $$
 teqp supports the virial coefficient directly, there is the ``get_B2vir`` method for the second virial coefficient:
 %% Cell type:code id:720e120c tags:
 ``` python
 model.get_B2vir(300, z)
 ```
 %% Output
    -0.00023661263734465424
 %% Cell type:markdown id:2fba4369 tags:
 And the ``get_Bnvir`` method that allows for the calculation of higher virial coefficients:
 %% Cell type:code id:92d01992 tags:
 ``` python
 model.get_Bnvir(7, 300, z)
 ```
 %% Output
    {2: -0.0002366126373446542,
     3: 3.001768410777936e-08,
     4: -3.2409760373816355e-12,
     5: 3.9617816466337214e-16,
     6: -4.552923983836698e-20,
     7: 5.3759278511184914e-24}
 %% Cell type:markdown id:d50998b1 tags:
 The ``get_Bnvir`` method was implemented because when doing automatic differentiation, all the intermediate derivatives are also retained.
 There is also a method to calculate temperature derivatives of a given virial coefficient
 %% Cell type:code id:fa4ff386 tags:
 ``` python
-model.get_dmBnvirdTm(2, 3, 300, z) # third temperature derivative of the
+model.get_dmBnvirdTm(2, 3, 300, z) # third temperature derivative of the second virial coefficient
 ```
 %% Output
    1.0095625628421257e-10