review derivation: first law of thermodynamics

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
9 multiply both sides by
1. 3464107376; locally 2714175:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
1. 5074423401:
$$V$$
$$pdg_{7586}$$
1. 6397683463; locally 7939101:
$$V \alpha = \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} pdg_{7586} = \frac{d}{d pdg_{7343}} pdg_{7586}$$
valid 3464107376:
6397683463:
3464107376:
6397683463:
6 substitute LHS of two expressions into expr
1. 1085150613; locally 4576755:
$$C_V = \left(\frac{\partial U}{\partial T}\right)_V$$
$$pdg_{6682} = \frac{d}{d pdg_{7343}} pdg_{5786}$$
2. 5634116660; locally 7384950:
$$\pi_T = \left(\frac{\partial U}{\partial V}\right)_T$$
$$pdg_{5480} = \frac{d}{d pdg_{7586}} pdg_{5786}$$
3. 9941599459; locally 2445123:
$$dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV$$
$$dU = \frac{d}{d pdg_{7343}} pdg_{5786}$$
1. 5002539602; locally 5358683:
$$dU = C_V dT + \pi_T dV$$
$$dU = dT pdg_{6682} + dV pdg_{5480}$$
failed 1085150613:
5634116660:
9941599459:
5002539602:
1085150613:
5634116660:
9941599459:
5002539602:
8 declare initial expr
1. 3464107376; locally 2714175:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 3464107376:
3464107376:
1 declare initial expr
1. 1815398659; locally 7368252:
$$U = Q + W$$
$$pdg_{5786} = pdg_{1088} + pdg_{9432}$$
no validation is available for declarations 1815398659:
1815398659:
7 divide both sides by
1. 5002539602; locally 5358683:
$$dU = C_V dT + \pi_T dV$$
$$dU = dT pdg_{6682} + dV pdg_{5480}$$
1. 8854422847:
$$dT$$
$$pdg_{7343}$$
1. 6055078815; locally 3830663:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
Nothing to split 5002539602:
6055078815:
5002539602:
6055078815:
11 simplify
1. 2257410739; locally 1136968:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
1. 7826132469; locally 1189259:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
Nothing to split 2257410739:
7826132469:
2257410739:
7826132469:
5 declare initial expr
1. 5634116660; locally 7384950:
$$\pi_T = \left(\frac{\partial U}{\partial V}\right)_T$$
$$pdg_{5480} = \frac{d}{d pdg_{7586}} pdg_{5786}$$
no validation is available for declarations 5634116660:
5634116660:
10 substitute LHS of expr 1 into expr 2
1. 6397683463; locally 7939101:
$$V \alpha = \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} pdg_{7586} = \frac{d}{d pdg_{7343}} pdg_{7586}$$
2. 6055078815; locally 3830663:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
1. 2257410739; locally 1136968:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
Nothing to split 6397683463:
6055078815:
2257410739:
6397683463:
6055078815:
2257410739:
2 declare initial expr
1. 9941599459; locally 2445123:
$$dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV$$
$$dU = \frac{d}{d pdg_{7343}} pdg_{5786}$$
no validation is available for declarations 9941599459:
9941599459:
hold volume constant in first term; hold temperature constant in second term
3 declare initial expr
1. 3547519267; locally 8155541:
$$S = k_{\rm Boltzmann} \ln \Omega$$
$$pdg_{1394} = pdg_{1157} \log{\left(pdg_{3434} \right)}$$
no validation is available for declarations 3547519267:
3547519267:
4 declare initial expr
1. 1085150613; locally 4576755:
$$C_V = \left(\frac{\partial U}{\partial T}\right)_V$$
$$pdg_{6682} = \frac{d}{d pdg_{7343}} pdg_{5786}$$
no validation is available for declarations 1085150613:
1085150613:
12 declare initial expr
1. 9781951738; locally 9670239:
$$\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$$
$$pdg_{4645} = - \frac{\frac{d}{d pdg_{8134}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 9781951738: error for dim with 9781951738
9781951738: N/A
Physics Derivation Graph: Steps for first law of thermodynamics

Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
7586 variable V
$$V$$
real
• length: 3
volume
15
7343 variable T
$$T$$
real
• temperature: 1
temperature
18
4645 variable \kappa_T
$$\kappa_T$$
real dimensionless coefficient of isothermal compressibility
• str_note
6
8134 variable P
$$P$$
real
• length: -1
• mass: 1
• time: -2
pressure
13
6682 variable C_V
$$C_V$$
real
• length: 2
• mass: 1
• temperature: -1
• time: -2
heat capacity at constant volume
2
5786 variable U
$$U$$
real
• length: 2
• mass: 1
• time: -2
internal energy
7
4686 variable \alpha
$$\alpha$$
real
• temperature: -1
expansion coefficient
6
9432 variable Q
$$Q$$
real
• length: 2
• mass: 1
• time: -2
heat flow
1
1394 variable S
$$S$$
real
• length: 2
• mass: 1
• temperature: -1
• time: -2
entropy
1
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
1088 variable W
$$W$$
real
• length: 2
• mass: 1
• time: -2
work done to a system
1
5480 variable \pi_T
$$\pi_T$$
real
• length: 1
• mass: 1
• time: -2
internal pressure at constant temperature
2
1157 constant k_{Boltzmann}
$$k_{Boltzmann}$$
['real']
• length: 2
• mass: 1
• temperature: -1
• time: -2
Boltzmann constant 1.38064852 10^{-23}   meter^2 kilogram second^-2 Kelvin^-1
1
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 2711162: 0