# 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 (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 |
specificheat ratio γ |
density |
internal energy |
enthalpy |

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 |
|||

Ṯ (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 |

# see Steam (properties)

### Notes

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

### Further Reading

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

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