Equating Heat and Einstein's equation confusion

[pjbeierle]

I should start by saying that I am a bit embarissed by asking such a silly question
By simply equating the mass-energy formula with the temperature dependence of heat...

M*c² = M* c_m *ΔT

it strikes me as odd that the mass cancels,

c² = c_m *ΔT

I was doing this in order to calculate how much mass is gained by heating say a cup of water up by a specific amount (I know it would be very small, but I was just curious of the order of magnitude). But it seems I cannot do this, so there must be something obvious I am missing.

 

[russ_watters]

They don't cancel because they are two different masses. On the left is the mass equivalent of the energy and on the right is the rest mass of the water.

 

[DrGreg]

They don't cancel because they are two different masses. On the left is the mass equivalent of the energy and on the right is the rest mass of the water.

...or to put it another way, the equation should be
$$\Delta m \, c^2 = m \, c_m \, \Delta T$$