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# Heat Capacity

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Short Summary:
The heat capacity is defined as: dT: temperature change
d'
Q: heat added to the system

Heat capacity (Constant Volume )
For constant volume, the heat capacity is defined as:

Cv = dU / dT

Heat capacity (Constant Pressure )
For constant pressure, the heat capacity is defined as:
Cp = d'Q / dT

The heat capacity of a body is the quantity of energy needed to cause its temperature to change by 1o C. The heat capacity, C, of a system is the ratio of the heat added to the system, or withdrawn from the system, to the resultant change in the temperature:

C = q/ΔT = δq/dT [J/deg]

The heat capacity of a body depends on what substance (s) it is made of and the masses of the different substances in the body. The specific heat capacity of a substance is the quantity of energy needed to change the temperature of 1 kg of substance by 1o C.

This definition is only valid in the absence of phase transitions.

Units of specific heat capacity are joule
kg -1 oC -1.

New state of the system is not defined by T only, need to specify or constrain the second variable: : heat capacity at constant volume : heat capacity at constant pressure

The fact that
δq is not a state function and depends on the path is reflected in the dependence of the heat capacity on the path, cp ≠ cv

(Note that small c is used for the derived intensive quantity, per mass, per volume, or per mole, versus capital C for the extensive quantity. For a system containing n moles Cp = ncp and Cv = ncv where cv and cp are molar values).

cv and cp can be measured experimentally

isobaric process: dH = δq =
cp dT
isochoric process: dU = δq =
cv dT

H and U can be calculated from
cp and cv

cv VS cp
If a material is allowed to expand during heating, how does this affect its heat capacity? Since U= U(V,T) Differentiation with respect to T at constant P gives:  : work of expansion at constant P due to the temperature increase by dT : work of expansion against internal cohesive forces due to the temperature increase by dT

Calculation of enthalpy from heat capacity
For P = constant and dH = cp dT , the integration gives: Example (1)
Let us find enthalpy for copper at 500K.
cp ≈ 24.4 Jmol-1K-1 for copper at 1atm.

From the first law we can only calculate the difference ΔH – need a reference enthalpy at 1atm and 298 K is called enthalpy of formation, H298. For pure elements in their equilibrium states H298 = 0. Enthalpy of substances other than pure elements can also be calculated.
The enthalpy of a compound at 298 K =
standard heat of formation of the substance from the elements.

Example (2)
For oxidation of copper at 25 oC: Cusolid + 1/2C2gas = CuOsolid The reaction is exothermic – heat and/or work are produced.

Example (3)
What is the specific heat capacity of water?
Ans: The specific heat capacity of water is Swater= 4200 joule kg -1 oC -1 (approximately)

Example (4) what are the two principle specific (or molar) heat capacities of gases?
Ans: Two principle specific (or molar) heat capacities of gases are:

The specific (or molar) heat capacity  at constant volume, Cv
The specific (or molar) heat capacity  at constant pressure, Cp

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