Except during phase transitions, substances undergo temperature change when heat is added or removed. In thermodynamics, the rate at which this temperature change occurs can be characterized using two properties: specific heat and heat capacity.

This article explores the differences between these two properties and how they are related.

## Specific Heat vs Heat Capacity

Substances have varying heat storage capabilities, depending on several factors such as their molecular composition and structure, the presence of impurities, the phase they exist in, and the local conditions surrounding them.

During phase transitions, a substanceโs ability to hold heat can be characterized using the latent heats of fusion, vaporization, and sublimation for the solid-liquid, liquid-gas, and solid-gas transition regions, respectively. Within the single-phase region, this can be characterized using the specific heat and heat capacity.

Both the specific heat and heat capacity quantify the thermal energy required to increase or decrease the temperature of a substance by a unit. However, the difference lies in the nature of these properties: specific heat is an intensive property, which means that it is independent of the amount of substance, while heat capacity is an extensive property, which means that its value is affected by the amount of substance or the size of the system.

Specific heat pertains to the amount of heat energy needed to raise or lower the temperature of a unit mass of a substance by one degree. On the other hand, heat capacity pertains to the amount of heat energy required to alter the temperature of the entire mass of substance or the whole system by one degree.

Given a specific heat, the required heat needed to increase or decrease the temperature of a substance can be calculated using the following formula:

Where:

- Q = heat required to increase or decrease the temperature by ฮT [J or Btu]
- m = mass of the substance [kg or lb
_{m}] - c = specific heat of the substance [J/kg-K or Btu/lb
_{m}-ยฐF] - ฮT = change in temperature [K or ยฐF]

On the other hand, given a heat capacity for a specific amount of substance, the required heat needed to increase or decrease its temperature can be calculated using the following formula:

Where:

- C = heat capacity of the substance [J/K or Btu/ยฐF]

Although heat is the most common form of energy associated with specific heat and heat capacity, it is important to note that the temperature of a substance can be changed by the transfer of energy in any form, not just thermal energy.

## Specific Heat and Heat Capacity Formula

Based on the above definition, specific heat can be derived from heat capacity using the following formula:

It is also common to express specific heat on a molar basis. In this case, the formula becomes:

Where:

- cฬ = molar specific heat [J/mol-K or Btu/mol-ยฐF]
- n = number of moles of the substance [mol]

It is important to note that, for compressible substances, the magnitudes of the specific heat and heat capacity depend on how the process of heat transfer is executed. In thermodynamics, there are two kinds of specific heats: specific heat at constant volume, denoted by c_{v}, and specific heat at constant pressure, denoted by c_{p}.

Specific heat at constant volume refers to the energy needed to raise the temperature of a substance’s unit mass by one degree while keeping the volume constant. On the other hand, specific heat at constant pressure represents the energy required to achieve the same temperature increase while maintaining the pressure. The diagram below illustrates this concept.

At constant pressure, the system is allowed to expand. Since this expansion requires additional energy to be supplied to the system, the specific heat at constant pressure is always greater than the specific heat at constant volume.

Formally, the specific heat at constant volume is defined as the change in internal energy with temperature at constant volume. Mathematically, this can be expressed as:

Where:

- c
_{v}= specific heat at constant volume [J/kg-K or Btu/lb_{m}-ยฐF] - u = specific internal energy [J/kg or Btu/lb
_{m}] - T = temperature [K or ยฐF]

On the other hand, the specific heat at constant pressure is formally defined as the change in enthalpy with temperature at constant pressure. Mathematically, this can be expressed as:

Where:

- c
_{p}= specific heat at constant pressure [J/kg-K or Btu/lb_{m}-ยฐF] - h = specific enthalpy [J/kg or Btu/lb
_{m}]

As shown in the formulas above, c_{v} and c_{p} depend on internal energy and enthalpy of the substance, respectively. Since internal energy and enthalpy vary with pressure and temperature, it follows that c_{v} and c_{p} also vary with pressure and temperature.

This means that the energy required to increase or decrease a substance’s temperature depends on the temperature and pressure of the system. However, this difference is usually not very large. In practical applications, it is common to assume a constant specific heat and heat capacity for a substance.

For ideal gases, however, the specific heat depends solely on temperature and not on pressure or volume. In this case, c_{v} and c_{p} can be related to the gas constant using the following formula:

Where:

- R = gas constant [J/kg-K or Btu/lb
_{m}-ยฐF]

Furthermore, there is another ideal gas property that relates c_{v} and c_{p}, called the specific heat ratio. The specific heat ratio can be calculated using the following formula:

Where:

- k = specific heat ratio [unitless]

The specific ratio also varies with temperature, but this variation is almost negligible. For monatomic gases, its value is essentially constant at 1.667. For diatomic gases, including air, the specific heat ratio is approximately equal to 1.4 at room temperature.

The table shows the ideal-gas specific heats as well as the specific heat ratios of common gases at different temperatures.

### Specific Heats of Air

### Specific Heats of Carbon Dioxide, CO2

### Specific Heats of Carbon Monoxide, CO

### Specific Heats of Hydrogen, H2

### Specific Heats of Nitrogen, N2

### Specific Heats of Oxygen, O2

Note that c_{p} and c_{v} are not equal only for compressible substances. For incompressible substances, like most solids and liquids, c_{p} and c_{v} have equal values.

The table below shows the specific heats of common incompressible substances at different temperatures.