## review derivation: coefficient of thermal expansion using the equation of state for an ideal gas

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.

Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Notes for this derivation:
https://notendur.hi.is/hj/EE2/HD1lausn.pdf

Options
Alternate views of this derivation:
Edit this content:

To edit a step, click on the number in the "Index" column in the table below

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
8 simplify
1. 6925244346; locally 7845152:
$$\alpha = \frac{PV}{T} \frac{1}{VP}$$
$$pdg_{4686} = \frac{pdg_{7586} pdg_{8134}}{pdg_{7343}}$$
1. 2472653783; locally 2491768:
$$\alpha = \frac{1}{T}$$
$$pdg_{4686} = \frac{1}{pdg_{7343}}$$
LHS diff is 0 RHS diff is (pdg7586*pdg8134 - 1)/pdg7343 6925244346:
2472653783:
6925244346:
2472653783:
5 declare initial expr
1. 8435841627; locally 5130250:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
no validation is available for declarations 8435841627:
8435841627:
2 declare initial expr
1. 3497828859; locally 5927974:
$$V = \frac{n R T}{P}$$
$$pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}$$
no validation is available for declarations 3497828859: inconsistent dimensions
3497828859: N/A
7 substitute RHS of expr 1 into expr 2
1. 2613006036; locally 8283443:
$$\frac{PV}{T} = nR$$
$$\frac{pdg_{7586} pdg_{8134}}{pdg_{7343}} = pdg_{2834} pdg_{8179}$$
2. 5962145508; locally 4600503:
$$\alpha = \frac{nR}{VP}$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179}}{pdg_{7586} pdg_{8134}}$$
1. 6925244346; locally 7845152:
$$\alpha = \frac{PV}{T} \frac{1}{VP}$$
$$pdg_{4686} = \frac{pdg_{7586} pdg_{8134}}{pdg_{7343}}$$
LHS diff is 0 RHS diff is (-pdg7586*pdg8134 + 1)/pdg7343 2613006036:
5962145508:
6925244346:
2613006036:
5962145508:
6925244346:
1 declare initial expr
1. 3464107376; locally 5888046:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 3464107376: error for dim with 3464107376
3464107376: N/A
4 simplify
1. 1311403394; locally 7236464:
$$\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179} \frac{d}{d pdg_{7343}} pdg_{7343}}{pdg_{7586} pdg_{8134}}$$
1. 5962145508; locally 4600503:
$$\alpha = \frac{nR}{VP}$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179}}{pdg_{7586} pdg_{8134}}$$
valid 1311403394: error for dim with 1311403394
5962145508:
1311403394: N/A
5962145508:
6 divide both sides by
1. 8435841627; locally 5130250:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
1. 7924842770:
$$T$$
$$pdg_{7343}$$
1. 2613006036; locally 8283443:
$$\frac{PV}{T} = nR$$
$$\frac{pdg_{7586} pdg_{8134}}{pdg_{7343}} = pdg_{2834} pdg_{8179}$$
valid 8435841627:
2613006036:
8435841627:
2613006036:
9 declare final expr
1. 2472653783; locally 2491768:
$$\alpha = \frac{1}{T}$$
$$pdg_{4686} = \frac{1}{pdg_{7343}}$$
no validation is available for declarations 2472653783:
2472653783:
3 substitute LHS of expr 1 into expr 2
1. 3497828859; locally 5927974:
$$V = \frac{n R T}{P}$$
$$pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}$$
2. 3464107376; locally 5888046:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
1. 1311403394; locally 7236464:
$$\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179} \frac{d}{d pdg_{7343}} pdg_{7343}}{pdg_{7586} pdg_{8134}}$$
LHS diff is 0 RHS diff is -pdg2834*pdg8179/(pdg7586*pdg8134) + 1/pdg7343 3497828859: inconsistent dimensions
3464107376: error for dim with 3464107376
1311403394: error for dim with 1311403394
3497828859: N/A
3464107376: N/A
1311403394: N/A
Physics Derivation Graph: Steps for coefficient of thermal expansion using the equation of state for an ideal gas

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
4686 variable \alpha
$$\alpha$$
real
• temperature: -1
expansion coefficient
6
8134 variable P
$$P$$
real
• length: -1
• mass: 1
• time: -2
pressure
13
8179 constant R
$$R$$
real
• amount of substance: -1
• length: 2
• mass: 1
• temperature: -1
• time: -2
ideal gas constant 8.31446261815324   J⋅K^{−1} mol^{−1}
8
7343 variable T
$$T$$
real
• temperature: 1
temperature
18
7586 variable V
$$V$$
real
• length: 3
volume
15
2834 variable n
$$n$$
real dimensionless amount of substance
8
MESSAGE:
• local variable 'all_df' referenced before assignment