Editing Specific heat capacity (section)

===Conservation of energy===
The absolute value of this quantity <math>U</math> is undefined, and (for the purposes of thermodynamics) the state of "zero internal energy" can be chosen arbitrarily.  However, by the [[law of conservation of energy]], any infinitesimal increase <math>M \, \mathrm{d}U</math> in the total internal energy <math>M U</math> must be matched by the net flow of heat energy <math>\mathrm{d}Q</math> into the sample, plus any net mechanical energy provided to it by  enclosure or surrounding medium on it.  The latter is <math>-P \, \mathrm{d}V</math>, where <math>\mathrm{d} V</math> is the change in the sample's volume in that infinitesimal step.<ref name=fein>Feynman, Richard, ''[[The Feynman Lectures on Physics]]'', Vol. 1, Ch. 45</ref>  Therefore

<math display="block">\mathrm{d}Q - P \, \mathrm{d} V = M \, \mathrm{d}U</math>

hence

<math display="block">\frac{\mathrm{d}Q}{M} - P \, \mathrm{d}\nu = \mathrm{d}U</math>

If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount <math>\mathrm{d}Q</math>, then the term <math>P \, \mathrm{d}\nu</math> is zero (no mechanical work is done).  Then, dividing by <math>\mathrm{d} T</math>,

<math display="block">\frac{\mathrm{d}Q}{M \, \mathrm{d}T} = \frac{\mathrm{d}U}{\mathrm{d}T}</math>

where <math>\mathrm{d}T</math> is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume <math>c_V</math> of the material.

For the heat capacity at constant pressure, it is useful to define the [[specific enthalpy]] of the system as the sum <math>h(T, P, \nu) = U(T, P, \nu) + P \nu</math>. An infinitesimal change in the specific enthalpy will then be

<math display="block">\mathrm{d}h = \mathrm{d}U + V \, \mathrm{d}P + P \, \mathrm{d}V</math>

therefore

<math display="block">\frac{\mathrm{d}Q}{M} + V \, \mathrm{d}P = \mathrm{d}h</math>

If the pressure is kept constant, the second term on the left-hand side is zero, and

<math display="block">\frac{\mathrm{d}Q}{M \, \mathrm{d}T} = \frac{\mathrm{d}h}{\mathrm{d}T}</math>

The left-hand side is the specific heat capacity at constant pressure <math>c_P</math> of the material.