Introduction

The heat capacity ratio is heat capacity at constant pressure (C_P) to heat capacity at constant volume (C_V). It is also known as the adiabatic index, the ratio of specific heat, or Laplace’s coefficient. It’s also known as the isentropic expansion factor, and it’s represented by 𝛾 (gamma) for an ideal gas or 𝜅 (kappa) for a real gas. Aerospace and chemical engineers both use the 𝛾 symbol.

where C is the heat capacity of gas, \bar{C} is its molar heat capacity (heat capacity per mole), and c is its specific heat capacity (heat capacity per unit mass). Constant-pressure and constant-volume situations are denoted by the suffixes P and V, respectively.

Table Of Contents

• Specific Heat Capacity
• Relationships with Ideal Gas
• Relationships with Real Gas
• Thermodynamic expressions
• Frequently Asked Questions (FAQs)

Specific Heat Capacity

The specific heat capacity (symbol c_p) of a substance is the heat capacity of a sample divided by the mass of the sample in thermodynamics. Specific heat capacity is also known as massic heat capacity. Unofficially, the quantity of heat must be added to one unit of mass of a substance to create a one-unit temperature increase. The joule per kelvin per kilogramme, or J.kg^-1. K^-1 is the SI unit for specific heat capacity. For example, it takes 4184 joules of heat to raise the temperature of 1 kilogramme of water by 1 K; hence water’s specific heat capacity is 4184 Jkg^-1K^-1.

When a substance, particularly a gas, is allowed to expand while heated (specific heat capacity at constant pressure), it may have a substantially larger specific heat capacity than when heated in a closed vessel that prohibits expansion (specific heat capacity at constant volume). The heat capacity ratio is the quotient

In the same way, specific gravity refers to the ratio between a substance’s specific heat capacities at a given temperature. For those of a reference substance at a reference temperature, such as water at 15 °C, the term specific heat can also refer to the ratio between the specific heat capacities at a given temperature and those of a known compound at a reference temperature.

Other intense metrics of heat capacity with different denominators are similarly connected to specific heat capacity. The molar heat capacity, whose SI unit is joule per kelvin per mole, Jmol^-1K^-1, is the result of measuring the amount of substance in moles. The volumetric heat capacity, whose SI unit is joule per kelvin per cubic metre, Jm^-3K^-1, is obtained by taking the amount to be the volume of the sample (as is sometimes done in engineering).

Relationships with Ideal Gas

The heat capacity of an ideal gas remains constant with temperature. As a result, we can write H = C_PT for the enthalpy and U = C_VT for the internal energy. As a result, the heat capacity ratio can also be defined as the ratio of enthalpy to internal energy:

Heat capacities can also be expressed in terms of the heat capacity ratio (𝜸) and the gas constant (R):

where n is the amount of substance in moles.

Mayer’s relation enables the value of C_V to be calculated from the more often tabulated value of C_P:

C_V = C_P – nR

Relationships with Real Gas

As the temperature rises, molecular gases get access to higher-energy rotational and vibrational states, increasing the degrees of freedom and decreasing. Both C_P and C_V grow with increasing temperature for a real gas while remaining separated by a fixed constant (as above, C_P = C_V + nR), reflecting the relatively constant PV difference in work done during expansions for constant pressure vs constant volume circumstances. As a result, as the temperature rises, the ratio of the two values, 𝜸 falls.

Thermodynamic expressions

Approximated values (especially C_P – C_V = nR) are frequently inadequately accurate for real engineering calculations, such as flow rates through pipes and valves. Wherever possible, an experimental value should be used rather than one based on this estimate. By estimating C_V from the residual characteristics, a rigorous value for the ratio CP / CV can also be calculated:

C_P values are readily available and documented; however, C_V values must be determined using relationships like these.

The approach described above is used to create formal expressions using state equations (such as Peng–Robinson) that closely match experimental values, eliminating the requirement for a database of ratios or C_V values. The finite-difference approximation can also be used to determine values.