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By Levi Clancy for Student Reader on

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Heat Capacity

Q = m c ΔT

Q = Heat needed to change the object's temperature by ΔT (in J or Cal)
m = Mass of substance (in kg)
ΔT = Temperature difference (in °C)
c = Specific heat of substance (in kcal kg-1 C-1 or J kg-1 C-1)

Note that since ΔT measures relative temperature, which is the same in °C and °K.
However, it is best to just use °C for the sake of consistency.

Specific Heat (c)J kg-1 C-1The specific heat c is a unique property of a given material. It represents the amount of heat needed to change 1 kilogram of a substance from 14.5°C to 15.5°C. While its units are officially J kg-1 C-1, also often used are kcal kg-1 C-1
Phase Changes

If a substance undergoes a phase change (for example, a cube of ice melts when placed in a cup of water) then the two phases and the phase change must all be accounted for:

m cice | 0 - Tice | + m cwater | 0 - Tfinal | + m Lliquefaction

Q = m L

Q = Amount of heat needed to change the phase of the object (in J or Cal)
m = Mass of substance
L = Latent heat of substance (like specific heat, in kcal kg-1 C-1 or J kg-1 C-1)

Latent Heat (L)J kg-1 C-1The latent heat L of a substance refers to the amount of energy ttransfer in its phase change; this is in addition to the heat normally involved in cooling or heating the substance.
Final Temperature (No Phase Changes)
Consider two different systems with different temperatures that come into contact with each other and where heat exchange is possible, such as hot coffee poured into a room temperature caraffe. Placed in contact, the two systems at different temperatures will equilibriate to a shared common temperature. The final temperature may be calculated from the relationship that the heat lost by one system is the heat gained by another:

m1 c1 | T - T1 | = m2 c2 (| T - T2 |

Where 1 refers to one system; 2 refers to the second system; and T refers to the final temperature.
Considering just two systems is simple, such as with the coffee and the caraffe. One side of the equation has the system losing heat; the other side of the equation has the system gaining heat. However, what if a third system is included? Simply determine whether it loses or gains heat, then add it to the appropriate side of the equation. Consider if hot coffee is poured into a room temperature cup with an insulating sleeve:

m1 c1 | T - T1 | = m2 c2 | T - T2 | + m3 c3 | T - T3 |

Where 1 refers to the system losing heat (coffee) and 2 and 3 refer to the systems gaining heat (cup and sleeve).
Final Temperature (Considering Phase Changes)
To what temperature will 5000 Joules of heat raise 1.0 kg of water that is initially at 20.0°C?
A small immersion heater is rated at 200 W. Estimate how long it will take to heat a 250 mL cup of water from 20°C to 80°C.
What is the specific heat of a metal substance if 100 kJ of heat is needed to raise 5 kg of the metal from 20°C to 40°C?
A 200 g lump of iron at 200°C is placed in a 100 g aluminum calorimeter cup containing 300 g of a cool fluid at 0°C. The final temperature is observed to be 30°C. Estimate the specific heat of the fluid.
A 20 gram ice cube at its melting point is dropped into an insulated container of liquid ethane. How much ethane evaporates if it is at its boiling point of 184.45°K and has a latent heat of vaporization of 488.76 kJ kg-1 C-1 (via Air Liquide)? Assume for simplicity that the specific heat of ice is a constant and is equal to its value near its melting point.