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Heat Capacity (specific and actual)

This page outlines the basics for establishing and using the various heat capacities for gases and vapours.

For the purposes of this page; substance refers to a gas or vapour

The 'specific' property of any substance refers to that property 'per unit value', which could be unit; mass, temperature, length, area, etc.
for example, density (kg/m³) is also specific mass (mass per unit volume), and
specific heat capacity refers to the energy that can be absorbed by a unit mass of a substance (at a given temperature).

Metric {units} are used throughout to minimise confusion. All units may be converted to Imperial using CalQlata's UniQon calculator

Constants

Boltzmann’s constant (KB) defines the quantity of heat energy required to raise the temperature of any particle (irrespective of the number or type of atom(s)) by 1 degree
KB = 1.38065156E-23 {J/K}

Avogadro defined the 'mole' as the quantity of any substance that contains the same number of particles as one gramme of Carbon-12 (¹²C), which is known as Avogradro’s number
NA = 6.02214129E+23

Multiply Boltzmann’s constant by Avogadro’s number and you have the heat energy required to raise the temperature of one mole of any substance by 1 degree, which is known as the universal gas constant (Rᵢ) for all ideal gases
Rᵢ = KB.NA = 8.314478767 {J/K/mol}

Specific Heat Capacity

Specific heat capacity {J/K/g} is the energy (potential and kinetic) that can be absorbed by a specific substance per degree (temperature) per unit mass

The gas constant for a unit mass of a specific gas (Rₐ) is the ideal gas constant (Rᵢ) divided by the relative atomic mass (RAM) of the gas molecule
Rₐ = Rᵢ/RAM {J/K/g}
RAM can be calculated using CalQlata's Elements calculator

Specific heat capacity in a constant temperature process is the gas constant (Rₐ)

Specific heat capacity in a constant volume process (cv) is the kinetic energy that can be absorbed by each gas microstate (N) and can be converted into work
(see Microstates below).
cv = N.Rₐ {J/K/g}

Specific heat capacity in a constant pressure process (cp) is the total specific heat capacity
cp = cv + Rₐ {J/K/g}

Heat Capacity (mass-specific)

Heat capacity is the amount of heat energy that can be absorbed by an actual mass of a particular substance. The above specific heat capacities may be converted into heat capacities by multiplying the specific heat capacity of a substance by its mass, as follows:
R = m.Rₐ {J/K}
Cv = m.cv {J/K}
Cp = m.cp {J/K}

Microstates

Microstates (N) are the translational, rotary and vibratory energy levels in each particle comprising the total mass of a substance governed by the relationship:
cp.Ln(Ṯ).RAM = KB.NA.Ln(N) {J/K/mol}

'N' varies with temperature, values provided are at 273.15K ...

In a constant temperature process: Nt = EXP[cp . Ln(Ṯ) / Rₐ]
Monatomic molecule (one atom): Nt = 1.5
Diatomic molecule (two atoms): Nt = 2.5
≥ Triatomic molecule (three atoms): Nt = 3.5

In a constant volume process: Nv = cv/Rₐ
Monatomic molecule (one atom): Nv = 1.5
Diatomic molecule (two atoms): Nv = 2.5
≥ Triatomic molecule (three atoms): Nv = 3.0

In a constant pressure process: Np = cp/Rₐ
Monatomic molecule (one atom): Np = 2.5
Diatomic molecule (two atoms): Np = 3.5
≥ Triatomic molecule (three atoms): Np = 4

Example Calculations

e.g. water vapour at 273.15K:
Rₐ = Rᵢ/RAM = 8.31447877 ÷ 18.02958 = 0.461157651 {J/K/g}

Water is a triatomic molecule (H₂O), so:

Microstate in a constant temperature process (Nt)
Using the actual value for cp and the following formula; N = EXP[cp . Ln(Ṯ) / Rₐ] = 3.5
where: cp = 1.856690184, Ṯ = 273.15K & Rₐ = 0.461157651 {J/K/g}
which is identical to the generally accepted value for Nt of 3.5 for steam @ 273.15K

Microstate in a constant volume process (Nv)
From cv = N.Rₐ and the known value for cv:
Nv = cv/Rₐ = 1.3955 ÷ 0.461157651 = 3.02615
which is close to the generally accepted value for Nv of 3.028 for steam @ 273.15K

Microstate in a constant pressure process (Np)
From cp = N.Rₐ and the known value for cp:
Np = cp/Rₐ = 1.856690184 ÷ 0.461157651 = 4.0262
which is close to the generally accepted value for Np of 4 for steam @ 273.15K

The difference between the above microstate values is due to the vibration energy being partially locked up or freed dependent upon process condition.

From just the specific heat capacity of your substance, using the above formulas you can calculate all the following additional properties of your gas or vapour
(@ atmospheric pressure = 1.0 bar = 1E+05 N/m²):

Process Independent Properties

temperature shc shc specific
heat ratio

γ
density internal energy enthalpy

(K)
cp
(J/K/g)
cv
(J/K/g)
ρ⁽¹⁾
(g/m³)
u
(J/g)
h
(J/g)
⁽²⁾cp - Rₐcp / cvp / Ṯ.Rₐ #Ṯ.cvu + Ṯ.Rₐ
1751.851.38881.33201239.118243.0474323.75
273.161.85671.39551.3305793.8409381.2037507.1735
373.151.88931.42811.3229581.1218532.9113704.9923
523.151.96791.50671.3061414.5788.25231029.507
1273.162.47282.01161.2292170.32082561.14263148.27
4082.5569183.22382.76261.166953.115111278.644613161.347
60003.352.88881.159636.140917333.054120100

Process Dependent Properties

temperature constant temperature constant volume constant pressure

(K)
Nt
s
(J/K/g)
Nv
s
(J/K/g)
Np
s
(J/K/g)
EXP(cp.Ln(Ṯ)/Rᵢ) Rₐ.Ln(Nt) cv/Rₐ Rₐ.Ln(Nv) cp/Rₐ Rₐ.Ln(Np)
175 3.1556 0.5300 3.0116 0.5084 4.0116 0.6406
273.16 3.5 0.5777 3.0262 0.5106 4.0262 0.6423
373.15 3.8407 0.6206 3.0969 0.5213 4.0969 0.6503
523.15 4.4 0.6833 3.2673 0.5460 4.2673 0.6691
1273.16 8.3834 0.9805 4.3622 0.6793 5.3622 0.7745
4082.556918 25.1234 1.4867 5.9907 0.8256 6.9907 0.8968
6000 33.286 1.6164 6.2643 0.8462 7.2643 0.9145

As can be seen from the above tables; microstate energy and entropy both increase with temperature (third law of thermodynamics).

Combining Specific Heats in Multiple Gases

The Gas constant (Rₐ) is the amount of energy required to raise the temperature of a unit mass of a substance by 1 degree.

Specific heat capacity (cp) is a term used to quantify the heat energy that can be absorbed by the unit mass of a substance at a given temperature in a constant pressure process.

Specific heat capacity (cv) is a term used to quantify the heat energy that can be absorbed by the unit mass of a substance at a given temperature in a constant volume process.

The lower the specific heat capacity of a substance the less heat energy the substance can absorb. Conversely, the higher its specific heat capacity the more heat energy the substance can absorb at the same temperature. The heat felt in, for example, the earth's atmosphere comes from the combined specific heat capacity (cv) of all its gases.

Each gas in an ideal gas mixture contributes to its equivalent specific heat capacity as follows:
cp = 1/m . ∑ mˀ . cpˀ
cv = 1/m . ∑ mˀ . cvˀ
R = 1/m . ∑ mˀ . Rˀ
where:
mˀ is the mass of each gas in the mixture
cpˀ is specific heat capacity of each gas in the mixture at constant pressure
cvˀ is specific heat capacity of each gas in the mixture at constant volume
Rˀ is the gas constant of each gas in the mixture
m is the mass of the gas mixture
cp is the specific heat capacity of the mixture at constant pressure
cv is the specific heat capacity of the mixture at constant volume
R is the gas constant of the mixture

Example Calculations

The gas constant and specific heats of 1kg of a gas mixture containing the following 3 gases:
Nitrogen (N₂) 78% by mass, cpᴺ = 983 J/kg/K, RAM = 14.0067 x 2
Oxygen (O₂) 20% by mass, cpᴼ = 919 J/kg/K, RAM = 15.9994 x 2
Argon (Ar) 1% by mass, cpᴬ = 531 J/kg/K, RAM = 39.948
... can be established as follows:

Gas Constant (R):

To calculate Rᴳ for each gas, you must first determine the number of moles of each gas in the mixture:
nˀ = mass(g) ÷ RAM(g/mol)

Nitrogen (N₂): mass = 780g & RAM = 28.0134g/mol
nᴺ = 780 g ÷ 28.0134 g/mol = 27.8438176 moles
Rᴺ = nᴺ.Rᵢ = 27.8438176 x 8.314479 = 231.5074214 J/K/mol

Oxygen (O₂): mass = 210g & RAM = 31.9988g/mol
nᴼ = 210 g ÷ 31.9988 g/mol = 6.250234384 moles
Rᴼ = nᴼ.Rᵢ = 6.250234384 x 8.314479 = 51.96757379 J/K/mol

Argon (Ar): mass = 10g & RAM = 39.948g/mol
nᴬ = 10 g ÷ 39.948 g/mol = 0.250325423 moles
Rᴬ = nᴬ.Rᵢ = 0.250325423 x 8.314479 = 2.08133073 J/K/mol

The gas constant for the mixture is the sum of the above:
Rᴳ = Rᴺ + Rᴼ + Rᴬ = 285.5556067 J/K/mol

Specific Heat Capacities:

Nitrogen (N₂): 0.78kg, cpᴺ = 983 J/kg/K, cvᴺ = 741.1 J/kg/K
Oxygen (O₂): 0.21kg, cpᴼ = 919 J/kg/K, cvᴼ = 657.3 J/kg/K
Argon (Ar): 0.01kg, cpᴬ = 531 J/kg/K, cvᴬ = 316.5 J/kg/K

cp = (0.78 x 983 + 0.21 x 919 + 0.01 x 531) ÷ 1 = 965.04 J/kg/K

cv = (0.78 x 741.1 + 0.21 x 657.3 + 0.01 x 316.5) ÷ 1 = 719.256 J/kg/K

γ = cp ÷ cv (ratio of specific heats)

Gas constants and specific heats for a number of pure gases are listed below:

Gas Rₐ cv cp γ
@ 273K J/kg/K J/kg/K J/kg/K
Air 286.8 678.6 965.4 1.423
Argon (Ar) 208.2 322.8 531 1.645
Butane (C₄H₁₀) 142.58 1511.2 1653.8 1.094
Carbon dioxide (CO₂) 188.85 655.15 844 1.288
Carbon monoxide (CO) 297 720.4 1017.4 1.412
Ethane (C₂H₆) 277.09 1339 1616.1 1.207
Ethylene (C₂H₄) 296.46 1377.7 1674.2 1.215
Helium (He₂) 2078.42 3161.6 5240 1.657
Hydrogen (H) 4125.63 10174 14300 1.405
Methane (CH₄) 518.7 1964.1 2482.8 1.264
Neon (Ne) 411.06 618.94 1030 1.664
Nitrogen (N₂) 297 686 983 1.433
Octane (C₈H₁₈) 72.85 1638.5 1711.3 1.044
Oxygen (O₂) 259.87 659.13 919 1.394
Propane (C₃H₈) 188.3 1457.1 1645.4 1.129
Water (15°C) 461.52372 3725.2763 4186.8 1.12389
Wet Steam (99.6°C) 461.52372 1400.0763 1861.6 1.3293
Dry Steam (450°C) # 461.52372 1633.7763 2095.3 1.2825
1 Btu/lb/°R = 1 cal/g/K = 4186.8 J/kg/K
# see Steam (properties)

Notes

  1. density is the reciprocal of specific volume (1/v)
  2. reference 34

Further Reading

You will find further reading on this subject in reference publications(1, 3, 12, 15 & 34)

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