# Heat Capacity & Transfer

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 (Nᵥ) and can be converted into work
(see Microstates below).
cᵥ = Nᵥ.Rₐ {J/K/g}

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

Rₐ = Rᵢ/RAM
Rᵢ = Rₐ.RAM
Nt = exp(Nᵨ.Ln(Ṯ))
Ln(Nt) = Nᵨ.Ln(Ṯ)
cᵨ.RAM.Ln(Ṯ) = KB.NA.Ln(Nt) = Rᵢ.Nᵨ.Ln(Ṯ)
cᵨ = cᵥ + Rₐ
cᵥ.RAM.Ln(Ṯ) + Rₐ.RAM.Ln(Ṯ) = Rₐ.Nᵨ.RAM.Ln(Ṯ)
cᵥ + Rₐ = Rₐ.Nᵨ
cᵥ = Rₐ.(Nᵨ - 1)

## Microstates

Microstates (N) are the energy states of atomic particles that are governed by the relationship:
cᵨ.Ln(Ṯ).RAM = KB.NA.Ln(Nt) {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: Nᵨ = cᵨ/Rₐ
Monatomic molecule (one atom): Nᵨ = 2.5
Diatomic molecule (two atoms): Nᵨ = 3.5
≥ Triatomic molecule (three atoms): Nᵨ = 4

## 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
Cᵥ = m.cᵥ
Cᵨ = m.cᵨ

## Heat Capacity (particle-specific)

Sub sub-atomic particles may be defined as follows:
Rₐ = 15156.3563034308 J/g/K
cᵥ = 22734.5344551462 J/g/K
cᵨ = 37890.8907585769 J/g/K
Nᵥ = 1.5
Nᵨ = 2.5
cᵥ = Rₐ.(2.5 - 1)
cᵥ = 1.5 x Rₐ

Property electron proton neutron units
mass 9.1093897E-028 1.67262164E-024 1.67262164E-024 g
RAM 0.000548580318390698 1.00727638277233 1.00727638277233 g/mol
R 1.38065156E-23 2.5350849503779E-20 2.5350849503779E-20 J/K
Cᵥ 2.07097734E-23 3.80262743E-20 3.80262743E-20 J/K
Cᵨ 3.4516289E-23 6.3377124E-20 6.3377124E-20 J/K

## 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 cᵨ and the following formula; Nt = EXP[cᵨ . Ln(Ṯ) / Rₐ] = 3.5
where: cᵨ = 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 (Nᵨ)
From cᵨ = N.Rₐ and the known value for cᵨ:
Nᵨ = cᵨ/Rₐ = 1.856690184 ÷ 0.461157651 = 4.0262
which is close to the generally accepted value for Nᵨ 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 density internal energy enthalpy specificheat ratioγ Ṯ(K) cp(J/K/g) cv(J/K/g) ρ⁽¹⁾(g/m³) u(J/g) h(J/g) ⁽²⁾ cp - Rₐ cp / cv p / Ṯ.Rₐ # Ṯ.cv u + Ṯ.Rₐ 175 1.85 1.3888 1.3320 1239.118 243.0474 323.75 273.16 1.8567 1.3955 1.3305 793.8409 381.2037 507.1735 373.15 1.8893 1.4281 1.3229 581.1218 532.9113 704.9923 523.15 1.9679 1.5067 1.3061 414.5 788.2523 1029.507 1273.16 2.4728 2.0116 1.2292 170.3208 2561.1426 3148.27 4082.556918 3.2238 2.7626 1.1669 53.1151 11278.6446 13161.347 6000 3.35 2.8888 1.1596 36.1409 17333.0541 20100

### 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 (cᵨ) 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:
cᵨ = 1/m . ∑ mˀ . cᵨˀ
cᵥ = 1/m . ∑ mˀ . cᵥˀ
R = 1/m . ∑ mˀ . Rˀ
where:
mˀ is the mass of each gas in the mixture
cᵨˀ 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
cᵨ 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, cᵨᴺ = 983 J/kg/K, RAM = 14.0067 x 2
Oxygen (O₂) 20% by mass, cᵨᴼ = 919 J/kg/K, RAM = 15.9994 x 2
Argon (Ar) 1% by mass, cᵨᴬ = 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, cᵨᴺ = 983 J/kg/K, cᵥᴺ = 741.1 J/kg/K
Oxygen (O₂): 0.21kg, cᵨᴼ = 919 J/kg/K, cᵥᴼ = 657.3 J/kg/K
Argon (Ar): 0.01kg, cᵨᴬ = 531 J/kg/K, cᵥᴬ = 316.5 J/kg/K

cᵨ = (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 = cᵨ ÷ cᵥ (ratio of specific heats)

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

Gas Rₐ cᵥ cᵨ 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)

All heat energy is electro-magneric radiation; i.e. radiated. Convection and conduction are simply different forms of radiation.

Convection: Heat is said to rise because atoms with the highest [kinetic] energy electrons will generate the greatest magnetic repulsion energy. The highest electro-static separation forces (highest temperature) will therefore balance furthest from the source of gravitational (magnetic) attraction.

Conduction: Heat passes through any solid or liquid as a result of the transfer of radiated electro-magnetic energy between proton-electron pairs in adjacent atoms.

Heat is absorbed by electrons, converted to electron velocity and radiated by an proton-electron-pair (an electron orbiting a proton)

For an electron at rest:
Rₐ = Rᵢ/RAM J/K/g
RAMₑ = 0.00054858031839 g/mol (RAM of an electron)
Rₐ = 15156.3563034308 J/K/g
Nᵥ = 1.5 & Nₚ = 2.5
cᵥ = Rₐ.Nᵥ = Rₐ.(Nₚ-1) = 22734.5344551462 J/K/g
cp = cᵥ+Rₐ = Rₐ.Nᵥ+Rₐ = Rₐ.(Nᵥ+1) = 37890.8907585769 J/g/K
Nᵥ = Nₚ -1 = Nₚ - Nₜ
Ṯ = exp(Rₐ.Ln(Nₜ)/ct) = 1
so @ rest; Nₜ = 1 & Ṯ = 1K

The relationship between electron velocity (v) and temperature (Ṯ) may be defined as; Ṯ = X.v² K
where: X = me² / mp.KB = 3.59332719743457E-11 K.s²/m²

The relationship between electron orbital radius (electro-magnetic amplitude) and temperature (Ṯ) may be defined as; Ṯ = XR/R K
where: XR = G.me² / φ.KB = 3.59332719743457E-11 K.s²/m²

It is interesting to note that Ṯ = (RAC.e.mₑ² / Rᵢ.mt) . v²/e² (from which the above factors [X & XR] have been derived) is similar to;
Newton's gravitational force F = G.m₁.m₂ / R²
Coulomb's force F = k.Q₁.Q₂ / R²
and Gilbert's and Maxwell's formulas for force and energy.
It is therefore anticipated that all of these formulas will eventually become just two; one for magnetism (gravity) and the other for electrical charge, or three; another for electro-magnetic energy.
Heat energy in the atom; Eₐ = cₚ.mₑ.Σn.Ṯₑ
where; 'ct' is the specific heat capacity of an electron, 'mₑ' is its mass & 'n' is the number of electrons at temperature 'Ṯₑ'
The [effective] temperature of the atom is; Ṯ = Eₐ / m.cp
where 'm' is the total mass of electrons in the atom

### Notes

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