Calculate True Temperature: Pyrometer Emissivity Correction

A pyrometer measures the temperature based on the radiant energy emitted by an object. This measurement depends on the object's surface emissivity ($\epsilon$). The radiant energy ($Q$) is proportional to $\epsilon T^4$, where $T$ is the absolute temperature (in Kelvin).

The pyrometer calculated a temperature ($T_{calc}$) of $1000\ ^\circ\text{C}$ assuming an emissivity ($\epsilon_{assumed}$) of $0.8$. The true emissivity ($\epsilon_{true}$) was later found to be $0.7$. We need to find the true temperature ($T_{true}$).

The fundamental relationship is based on equating the radiant energy flux:

Step-by-Step Calculation

1. Convert Calculated Temperature to Kelvin:

   The pyrometer's calculated temperature must first be converted from Celsius to Kelvin for the calculation.

   $T_{calc, K} = 1000\ ^\circ\text{C} + 273.15 = 1273.15\ \text{K}$
2. Calculate True Temperature in Kelvin:

   Rearrange the energy flux equation to solve for the true temperature in Kelvin ($T_{true, K}$):

   $T_{true, K}^4 = T_{calc, K}^4 \times \frac{\epsilon_{assumed}}{\epsilon_{true}}$ $T_{true, K} = T_{calc, K} \times \left( \frac{\epsilon_{assumed}}{\epsilon_{true}} \right)^{1/4}$

   Substitute the known values:

   $T_{true, K} = 1273.15\ \text{K} \times \left( \frac{0.8}{0.7} \right)^{1/4}$ $T_{true, K} = 1273.15\ \text{K} \times (1.142857)^{1/4}$ $T_{true, K} \approx 1273.15\ \text{K} \times 1.03285$ $T_{true, K} \approx 1315.44\ \text{K}$
3. Convert True Temperature back to Celsius:

   Convert the calculated true temperature from Kelvin back to degrees Celsius.

   $T_{true, ^\circ\text{C}} = T_{true, K} - 273.15$ $T_{true, ^\circ\text{C}} = 1315.44\ \text{K} - 273.15$ $T_{true, ^\circ\text{C}} \approx 1042.29\ ^\circ\text{C}$

The calculated actual temperature is approximately $1042.29\ ^\circ\text{C}$. Comparing this value to the options provided, $1043\ ^\circ\text{C}$ is the nearest value.