This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.
Notes for this derivation:
https://www.youtube.com/watch?v=QTiqFHtkS0 and https://www.youtube.com/watch?v=3Ylst3B49U
Clicking on the step index will take you to the page where you can edit that step.
Index  Inference Rule  Input latex  Feeds latex  Output latex  step validity  dimension check  unit check  notes 

9  multiply both sides by 



valid 
3464107376:
6397683463: 
3464107376:
6397683463: 

6  substitute LHS of two expressions into expr 



failed 
1085150613:
5634116660: 9941599459: 5002539602: 
1085150613:
5634116660: 9941599459: 5002539602: 

8  declare initial expr 



no validation is available for declarations 
3464107376:

3464107376:


1  declare initial expr 



no validation is available for declarations 
1815398659:

1815398659:


7  divide both sides by 



Nothing to split 
5002539602:
6055078815: 
5002539602:
6055078815: 

11  simplify 



Nothing to split 
2257410739:
7826132469: 
2257410739:
7826132469: 

5  declare initial expr 



no validation is available for declarations 
5634116660:

5634116660:


10  substitute LHS of expr 1 into expr 2 



Nothing to split 
6397683463:
6055078815: 2257410739: 
6397683463:
6055078815: 2257410739: 

2  declare initial expr 



no validation is available for declarations 
9941599459:

9941599459:

hold volume constant in first term; hold temperature constant in second term 
3  declare initial expr 



no validation is available for declarations 
3547519267:

3547519267:


4  declare initial expr 



no validation is available for declarations 
1085150613:

1085150613:


12  declare initial expr 



no validation is available for declarations 
9781951738:
error for dim with 9781951738 
9781951738:
N/A 
symbol ID  category  latex  scope  dimension  name  value  Used in derivations  references  

5480  variable 
\pi_T
\(\pi_T\) 
real 

internal pressure at constant temperature  2  
7586  variable 
V
\(V\) 
real 

volume  15  
5786  variable 
U
\(U\) 
real 

internal energy  7  
1088  variable 
W
\(W\) 
real 

work done to a system  1  
7343  variable 
T
\(T\) 
real 

temperature  18  
9432  variable 
Q
\(Q\) 
real 

heat flow  1  
1157  constant 
k_{Boltzmann}
\(k_{Boltzmann}\) 
['real'] 

Boltzmann constant 
1.38064852 10^{23} meter^2 kilogram second^2 Kelvin^1 
1  
4645  variable 
\kappa_T
\(\kappa_T\) 
real  dimensionless  coefficient of isothermal compressibility 

6  
1394  variable 
S
\(S\) 
real 

entropy  1  
8134  variable 
P
\(P\) 
real 

pressure  13  
3434  variable 
\Omega
\(\Omega\) 
real  dimensionless  number of microscopic configurations (known as microstates) that are consistent with the macroscopic quantities that characterize the system  1  
4686  variable 
\alpha
\(\alpha\) 
real 

expansion coefficient  6  
6682  variable 
C_V
\(C_V\) 
real 

heat capacity at constant volume  2 