Use the whole heat-capacity curve

A 0.200 kg sample remains in one phase while warming through 40 K. Its specific heat is c(θ) = 500 + θ J/(kg·K), where θ is the numerical rise in kelvins from its initial temperature. Work and heat losses are negligible.

How much heat is required?

Organize the givens & assumptions

Mass = 0.200 kg
Rise = 40 K
c changes linearly from 500 to 540 J/(kg·K)

Ideal insulated calorimetry at ordinary atmospheric pressure, no evaporation or other work, equilibrium final state and stated constant model capacities unless a temperature dependence is specified. Vessel heat capacity is negligible except where it is an explicit unknown. Pure ice/water melts at 0°C; no supercooling.

A $20800\ \mathrm J$
B $4160\ \mathrm J$
C $4000\ \mathrm J$
D $4320\ \mathrm J$

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