Materials engineers measured the electrical conductivity (\sigma, in 10^6\text{ S/m}) of a… — ACT

Materials engineers measured the electrical conductivity ( $\sigma$ , in $10^6\text{ S/m}$ ) of a copper-nickel alloy at various temperatures ( $T$ , in K). The measured values are recorded in the table below:

Temperature ( $T$ , K)	Electrical Conductivity ( $\sigma$ , $10^6\text{ S/m}$ )
300	14.2
400	11.8
500	9.4
600	7.0
700	4.6

Based on the data, if the linear relationship between temperature and electrical conductivity continues beyond $700\text{ K}$ , what is the predicted electrical conductivity of the alloy at $800\text{ K}$ ?

$2.2 \times 10^6\text{ S/m}$ Answer
B
$3.4 \times 10^6\text{ S/m}$
C
$4.6 \times 10^6\text{ S/m}$
D
$7.0 \times 10^6\text{ S/m}$

Answer

The predicted electrical conductivity at

800\text{ K}

2.2 \times 10^6\text{ S/m}

The electrical conductivity decreases by a constant rate of

2.4 \times 10^6\text{ S/m}

for every

100\text{ K}

increase in temperature. At

700\text{ K}

, the conductivity is

4.6 \times 10^6\text{ S/m}

. Extrapolating to

800\text{ K}

requires subtracting

2.4 \times 10^6\text{ S/m}

from

4.6 \times 10^6\text{ S/m}

, giving

2.2 \times 10^6\text{ S/m}

Step-by-Step Solution

Determine the change in electrical conductivity per

100\text{ K}

temperature increase.

Between

300\text{ K}

and

400\text{ K}

, conductivity decreases by

14.2 - 11.8 = 2.4 \times 10^6\text{ S/m}

. Checking other intervals confirms a constant rate of

-2.4 \times 10^6\text{ S/m}

per

100\text{ K}

Establishing the linear rate of change is necessary for linear extrapolation.

Apply the linear trend to extrapolate from

700\text{ K}

800\text{ K}

4.6 - 2.4 = 2.2 \times 10^6\text{ S/m}

Extrapolation involves extending the observed trend by one temperature interval (

100\text{ K}

) beyond the measured range.

Key Concept

Linear Extrapolation

Estimated Time:1m 0s

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