Final answer:
The energy, Q, required to raise the temperature of 1 kg of metal from 20 ℃ to 60 ℃ with a specific heat capacity that varies with temperature is calculated by integrating the function c(T) = 0.25 + 0.08T. The result is 138 Joules.
Step-by-step explanation:
To calculate the energy (Q) required to raise the temperature of 1 kg of metal from 20 ℃ to 60 ℃, we need to integrate the heat capacity function c(T) = a + bT over the temperature range. Given that a = 0.25 J/kg ℃ and b = .08 J/kg ℃^2, the integral of c(T) with respect to T will give us Q:
Q = ∫_{20}^{60} (a + bT) dT
Replacing a and b with their given values and integrating from 20 ℃ to 60 ℃:
Q = ∫_{20}^{60} (0.25 + 0.08T) dT
Q = [0.25T + 0.04T^2]_{20}^{60}
Calculating the definite integral, we get:
Q = (0.25 × 60 + 0.04 × 60^2) - (0.25 × 20 + 0.04 × 20^2)
Q = (15 + 144) - (5 + 16)
Q = 159 - 21 = 138 J
Therefore, the energy required to raise the temperature of the metal by 40 ℃ is 138 Joules.