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· Heat Capacity
· Thermodynamics - Introduction
· Van der Waals' Equation
· Clausius–Clapeyron Relation
· Gibbs Free Energy

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· Thermodynamics - Introduction
· First Law of Thermodynamics
· Ideal Gas Law
· Van der Waals' Equation
· Clausius–Clapeyron Relation
· Density
· Molar Mass
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· Helmholtz Free Energy
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Current Location > Physics Formulas > Thermodynamics > Heat Capacity

Heat Capacity

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:
C_v = dU / dT

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

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 The heat capacity, C, of a system is the ratio of the heat added to the system, or withdrawn from the system, to the resultant change in the temperature: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

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

Units of specific heat capacity are joule kg ^{-1 o}C ^-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 C_p = nc_p and C_v = nc_vwhere c_v and c_p are molar values).

c_v and c_p can be measured experimentally

isobaric process: dH = δq = c_p dT
isochoric process: dU = δq = c_v dT

H and U can be calculated from c_p and c_v

c_v VS c_p
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 = c_p dT , the integration gives:

Example (1)
Let us find enthalpy for copper at 500K.
c_p ≈ 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: Cu^solid + 1/2C₂gas = CuO^solid

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 S_water= 4200 joule kg ^{-1 o}C ^-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, C_v
The specific (or molar) heat capacity at constant pressure, C_p