Proving Equivalence of Equations with Constant Specific Heat

How can we prove that the two relations for changes in s, Eqs. 6.16 and 6.17, are equivalent once we assume constant specific heat?

Explanation:

To prove that equations 6.16 and 6.17 are equivalent when constant specific heat is assumed, we first have to recall the concept of specific heat, according to equation 3.27. Specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius. Therefore, when it is constant, it means that the energy required to raise the temperature remains the same. Now, let's consider equation 6.16, which might be a formula related to heat transfer (like Q=mcΔT) where Q is the heat energy transferred, m is the mass, and ΔT is the temperature change. In 6.17, given the context, this might be an energy balance equation. When we have constant specific heat, the two equations become equivalent because the specific heat 'c' in equation 6.16 is constant and does not change with temperature. Hence, no matter how much heat is transferred, dependent on mass and temperature difference, the 'specific heat' remains constant in both these equations rendering them equivalent. When constant specific heat is assumed, the two equations become equivalent because the energy required to raise the temperature remains the same; this constant factor is included in both equations. Therefore, despite differences in their structures, the constants make them equal.

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