Thermocouple laws and emf calculation

A thermocouple produces an emf when two dissimilar metals joined at two junctions are at different temperatures (Seebeck effect).

Laws of thermocouples

1. Law of homogeneous circuit: a thermoelectric current cannot be sustained in a circuit of a single homogeneous metal by heat alone, however the section varies. A temperature gradient in a uniform wire adds no emf.
2. Law of intermediate metals: a third metal inserted in the circuit (e.g. the voltmeter leads) adds no net emf if its two junctions are at the same temperature.
3. Law of intermediate temperatures: if the emf between junction temperatures $T_1$ and $T_2$ is $E_{1,2}$ and between $T_2$ and $T_3$ is $E_{2,3}$, then $E_{1,3} = E_{1,2} + E_{2,3}$. This lets tables with a 0 $^\circ$C reference be used for any cold junction.

(a) Junctions at 300 $^\circ$C and 100 $^\circ$C

$$E_{100,300} = E_{0,300} - E_{0,100} = 12.209 - 4.096 = 8.113\ \text{mV}$$

(b) Hot junction temperature

The instrument reads 7.5 mV with cold junction at 30 $^\circ$C. Referred to 0 $^\circ$C:

$$E_{0,T} = E_{30,T} + E_{0,30} = 7.5 + 1.203 = 8.703\ \text{mV}$$

This lies between 200 $^\circ$C (8.138 mV) and 300 $^\circ$C (12.209 mV).

$$T = 200 + \frac{8.703 - 8.138}{12.209 - 8.138}\times 100 = 200 + \frac{0.565}{4.071}\times 100 = 213.9\ ^\circ\text{C}$$

Answer: (a) 8.113 mV; (b) $T \approx 213.9\ ^\circ$C.