The water-gas reaction needs very high temperatures. The water-gas shift reac tion, aided by a catalyst, can operate effectively at lower temperatures.
The water-gas reaction requires high temperatures for the reaction to proceed effectively:
Reaction Temperature: \( T_1 \approx 1270 \, \text{K} \)
The water-gas shift reaction is typically carried out at lower temperatures in the presence of a catalyst:
Reaction Temperature: \( T_2 \approx 673 \, \text{K} \)
The temperature for the water-gas reaction (\( T_1 \)) is significantly higher than the temperature for the water-gas shift reaction (\( T_2 \)). Therefore:
\( T_1 > T_2 \)
Let \( \alpha = \dfrac{-1 + i\sqrt{3}}{2} \) and \( \beta = \dfrac{-1 - i\sqrt{3}}{2} \), where \( i = \sqrt{-1} \). If
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of \( m \) is ___________.