When a body gains or loses energy in a thermal process, we find experimentally that the temperature change of the body is proportional to its change in energy: ΔH = mS(ΔT)
where
Consider two bodies at different temperatures, insulated from all other bodies, yet able to transfer heat to each other. The heat lost by the initially hotter one is equal to the heat gained by the initially cooler one. Then the above equation applied to each body gives:
m1S1ΔT1 = m2S2ΔT2
HEAT: When an amount of energy is transferred from one body to another solely as a result of the temperature differences between the bodies, we call that amount of energy “heat”. Bodies can transfer energy in this way through three processes: radiation, conduction, convection. When the hotter body losses thermal energy to a cooler one, the process continues until both bodies reach the same “equilibrium” temperature.
NEWTON’S LAW OF COOLING: If the temperature difference ΔT, between the two bodies is not too large, the rate of change of the temperature difference is nearly proportional to that
temperature difference:
(d/dt)ΔT = -KΔT ...... (1)
If at some time t = 0 the temperature difference is T0 at a later time
t, the temperature difference will be:
ΔT = ΔT0 e-Kt ...... (2)
This can be shown by integrating equation (1).
Therefore, when two bodies exchange energy thermally the temperature of each will exponentially approach their common final equilibrium temperature.
HEAT UNITS: The kilocalorie(kc) is defined to be the amount of heat required to increase the temperature of 1kilogram of water through 1°C. In nutrition studies this is simply called the “calorie”. In physics books which use the cgs system the gram calorie (gc) or small calorie is defined as the amount of heat required to increase the temperature of 1gm of water by 1°C.
The British thermal unit (BTU) is defined as the amount of heat required to increase the temperature of 1pound of water by 1°F. In this system the specific heat capacity of water has magnitude 1.
CALORIMETRY: Consider two bodies at different temperatures, insulated from all other bodies, yet able to transfer heat to each other. The heat gained by the initially cooler body is equal to the heat lost by the initially hotter one.
m1s1 (ΔT)1 = m2 s2 (ΔT)2
Lead shots are heated to steam temperature, then dumped into cool water in the calorimeter and allowed to come to an equilibrium temperature. The necessary measurements are: mass of water, mw; initial water temperature, Tw; mass of metal sample, ms; sample temperature, Ts; and the final equilibrium temperature of the mixture, Tf. The heat exchange equation is:
mw(Tw - Tf) + msss(Ts - Tf) + mcsc(Tc - Tf ) = 0
The subscript c labels quantities that describe the calorimeter inner cup. This cup also participates in the heat exchange. Its temperature is nearly the same as the water in it, so Tc = Tw.
mcsc(Tw - Tf) + mtst(Tw - Tf)
Since these bodies follow the water temperature, they share the (Tw - Tf) factor with the water term of the heat transfer equation. We can therefore combine all three terms:
(mw + mcsc + mtst) (Tw - Tf)
The quantity (mssc + mtst) = meq is often called the "water equivalent" of the cup and thermometer. Once it is determined, this number is simply added to the measured mass of water, for the purposes of calculation of the heat transfer equation:
(mw + meq) (T2 - Tf) + msss (Ts - Tf) = 0
WATER EQUIVALENT: For a given volume of material, the effectiveness of the material for producing a temperature change is proportional to its water equivalent, the product of its mass and its specific heat capacity.
Equation for Error analysis:
$$ \frac{\Delta S_s}{S_s} = \frac{\Delta (T_c + T_f)}{(T_c + T_f)} + \frac{\Delta m_w}{m_w} + \frac{\Delta m_c}{m_c} + \frac{\Delta m_s}{m_s} + \frac{\Delta (T_s + T_f)}{(T_s + T_f)} $$