# Specific Heat and Latent Heat

- Specific heat: = energy required to change a unit mass of a material by 1°C. Units: energy per unit mass per degree.
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- Latent heat = energy required to change the state (gas, liquid, solid) of a unit mass of material. Units: energy per unit mass.

Both specific heat and latent heat are properties of a given material. In other words, every time the material is heated/cooled, no matter how quickly or by what heating process, the same amount of heat is transferred to reach the same state.

## Change in energy of a substance:

To raise/lower its temperature:

DE = C_{p}M DT
To change its state:

DE = LM

where

DE = change in energy

DT = temperature change

M = mass

C_{p} = specific heat

L = latent heat

Energy must be added

- to increase the temperature
- to melt it

- to evaporate it

Energy is released when a substance
Energy units:

- Ergs (1 erg = 1 g cm
^{2} s^{-2})
- Joules (1 J = 1 kg m
^{2} s^{-2} = 10^{7} ergs)
- Calories (1 cal = 4.184 J)

## Total energy to raise temperature of 1.0 kg of water from 10°C to 110°C

C_{p}^{water} = 4.2 x 10^{3} J kg^{-1} °C^{-1}

C_{p}^{steam} = 2.0 x 10^{3} J kg^{-1} °C^{-1}

L_{vaporization} = 2.3 x 10^{6} J kg^{-1}

M = 1 kg
DE = energy to heat water to boiling point +

energy to change state +

energy to raise temperature of steam

= C_{p}^{water} MDT +
L_{vaporization}M +
C_{p}^{steam}MDT

= (4.2 x 10^{3} J kg^{-1} °C^{-1})(1 kg)(90°C) +
(2.3 x 10^{6} J kg^{-1})(1 kg) +
(2.0 x 10^{3} J kg-1 °C^{-1})(1 kg)(10°C)

= 2.7 x 10^{6} J

## Total energy released at midocean ridges by
formation of new oceanic crust

C_{p}^{basalt} = 1.4 x 10^{3} J kg^{-1} °C^{-1}

C_{p}^{magma} = 1.0 x 10^{3} J kg^{-1} °C^{-1}

L_{melting} = 4.0 x 10^{5} J kg^{-1}

Temperature of magma when erupted = 1300°C

Melting point 1200°C

Temperature of sea water = 0°C

Length of ridge = 60,000 km = 6 x 10^{7} m

Crustal thickness = 5 km = 5 x 10^{3} m

Average spreading rate 2 cm yr^{-1} = 2 x 10^{-2} m yr^{-1}

Density of basalt 3 x 10^{3} kg m^{-3}

1. Mass of magma solidified (per year) = volume x density

= ridge length x crustal thickness * spreading rate* density

= (6 x 10^{7} m)( 5 x 10^{3} m)( 2 x 10^{-2 }m yr^{-1})( 3 x 10^{3} kg m^{-3})

=2 x 10^{13} kg yr^{-1}

2. DE = energy to solidify and cool crust (per year)

= energy to cool magma to melting point +

energy to solidify magma +

energy to cool basalt to ocean water temperature

DE = C_{p}^{magma} MDT +

L_{melting}M +

C_{p}^{basalt}MDT

= (1.0 x 10^{3} J kg^{-1} °C^{-1})(2 x 10^{13} kg)(100°C) +

(4.0 x 10^{5} J kg^{-1})(2 x 10^{13} kg) +

(1.4 x 10^{3} J kg^{-1} °C^{-1})(2 x 10^{13} kg)(1200°C)

= 2 x 10^{18} J + 8 x 10^{18} J + 3 x 10^{19} J

= (4 x 10^{19} J)(10^{7}ergs/ J)= 4 x 10^{26} ergs