Heat Capacity: Specific Heat, Cp, Cv and Thermodynamics

Heat capacity is a fundamental concept in thermodynamics that describes how much energy a substance absorbs or releases when its temperature changes. It helps explain why different materials warm up and cool down at different rates, even when they receive the same amount of heat. Heat capacity is important in chemistry, physics, engineering, climate science, and the design of thermal systems.

There are several related quantities: heat capacity, specific heat capacity, and molar heat capacity. For gases, heat capacity also depends on the experimental condition. The molar heat capacity at constant pressure is represented by \(C_p\), while the molar heat capacity at constant volume is represented by \(C_v\). Understanding their difference leads to the important ideal-gas relation \(C_p-C_v=R\).

What Is Heat Capacity?

Heat capacity is the amount of heat required to raise the temperature of an entire object or sample by one kelvin. For a small temperature change, its average value can be expressed as:

Here, \(C\) is heat capacity, \(\Delta Q\) is the heat transferred to the sample, and \(\Delta T\) is the temperature change. The SI unit of heat capacity is joules per kelvin (J/K). For a temperature-dependent material or a process with changing properties, the more precise differential definition is \(C=\delta Q/dT\) for the specified process.

Heat capacity is an extensive property: it generally increases with the amount of material. For example, a large pot of water has a greater total heat capacity than a small cup of water at the same conditions. This does not mean that the water itself has a different specific heat capacity.

Specific Heat Capacity

Specific heat capacity is the heat capacity per unit mass. It describes the energy required to raise the temperature of one unit of mass of a substance by one kelvin. It is commonly represented by \(c\) or \(s\).

In these equations, \(m\) is the mass of the material. The specific heat capacity can be measured in J/(kg·K) or J/(g·K). When mass is measured in grams and specific heat capacity in J/(g·K), the calculated heat is in joules.

Liquid water has a specific heat capacity of approximately \(4184\ \text{J/(kg·K)}\), or \(4.184\ \text{J/(g·K)}\), near ordinary room temperature. This relatively high value explains why water can absorb substantial thermal energy without a very large temperature increase.

Worked Example: Specific Heat Capacity

A 250 g sample absorbs 5,000 J of heat and its temperature increases by 10°C. Calculate its specific heat capacity.

The specific heat capacity is \(2.0\ \text{J/(g·K)}\). This result assumes that no phase change occurs and that the measured heat is absorbed by the sample rather than lost to the surroundings.

Molar Heat Capacity

Molar heat capacity is the heat required to raise the temperature of one mole of a substance by one kelvin under a specified condition. It is useful when comparing substances based on the number of particles rather than their mass.

Here, \(n\) is the amount of substance in moles, and \(C_m\) is the molar heat capacity in J/(mol·K). For gases, the molar heat capacity is commonly specified at constant pressure or constant volume.

Heat capacity depends on the process because a gas can expand or contract as it is heated. Consequently, \(C_p\) and \(C_v\) are generally different for gases. For solids and liquids, heat capacities are also defined under specified conditions, but the difference between pressure and volume constraints is often much smaller in ordinary laboratory situations.

Molar Heat Capacity at Constant Pressure (Cp)

The molar heat capacity at constant pressure, \(C_p\), describes the heat required per mole for a temperature increase at constant pressure. For a simple compressible system undergoing a constant-pressure process with only pressure-volume work, the heat supplied is related to the change in enthalpy.

Enthalpy is defined as:

where \(H\) is enthalpy, \(U\) is internal energy, \(P\) is pressure, and \(V\) is volume. At constant pressure, differentiating this expression gives:

For a reversible or quasi-static process involving only pressure-volume work, the first law can be written as \(\delta Q=dU+P\,dV\). Thus, at constant pressure, \(\delta Q=dH\). The molar heat capacity at constant pressure is therefore:

Here, \(H_m\) is molar enthalpy. The partial derivative indicates that pressure is held constant while temperature changes. This relation is particularly useful for interpreting heating experiments conducted at atmospheric pressure.

Molar Heat Capacity at Constant Volume (Cv)

The molar heat capacity at constant volume, \(C_v\), describes the heat required per mole to raise temperature while volume remains constant. When volume is fixed, pressure-volume work is zero because \(dV=0\). If no other form of work is involved, heat supplied increases the internal energy of the system.

For an ideal gas, internal energy depends only on temperature. Therefore, \(C_v\) is especially useful when calculating the internal-energy change of an ideal gas during a temperature change.

Derivation of Cp − Cv = R for an Ideal Gas

The relation \(C_p-C_v=R\) is known as Mayer’s relation. It applies to an ideal gas when \(C_p\) and \(C_v\) are molar heat capacities evaluated at the same temperature. The derivation uses the first law of thermodynamics and the ideal gas equation.

Why Is Cp Greater Than Cv?

When a gas is heated at constant volume, it cannot expand, so it performs no pressure-volume work. With only pressure-volume work possible, the supplied heat increases its internal energy.

When the same gas is heated at constant pressure, its volume increases. Some of the supplied energy is used to perform expansion work on the surroundings, in addition to increasing internal energy. More heat is therefore needed for the same temperature rise at constant pressure than at constant volume.

For an ideal gas, their difference is exactly \(R\) when expressed as molar heat capacities. For real gases, heat capacities and their difference can depend on temperature and pressure, so the ideal-gas relation may not hold exactly.

Ratio of Heat Capacities (Gamma)

The ratio of molar heat capacity at constant pressure to molar heat capacity at constant volume is called the ratio of heat capacities. It is represented by the Greek letter gamma, \(\gamma\), and has no units.

For an ideal gas with constant heat capacities, gamma appears in the equation for a reversible adiabatic process:

An adiabatic process has no heat transfer into or out of the system. The displayed pressure-volume relation assumes an ideal gas, a reversible adiabatic change, and constant heat capacities over the temperature range.

Typical Heat Capacity Values for Common Gases

The following are approximate molar heat capacities near 298 K. Gas heat capacities can change with temperature, and values for polyatomic gases are particularly sensitive to molecular motion.

Monatomic ideal gases have three translational degrees of freedom, giving \(C_v=\frac{3}{2}R\) and \(C_p=\frac{5}{2}R\). Their ratio is \(\gamma=\frac{5}{3}\). For a simple diatomic ideal gas at moderate temperatures, rotational modes contribute, giving approximately \(C_v=\frac{5}{2}R\), \(C_p=\frac{7}{2}R\), and \(\gamma=\frac{7}{5}\). Vibrational modes and non-ideal behaviour can alter these values.

Specific Heat Capacity vs. Molar Heat Capacity

Interactive Calculator: Cp, Cv and Gamma

Enter molar heat capacities to calculate gamma and compare the difference between Cp and Cv with the ideal-gas constant R. This calculator checks the numerical relation; it cannot determine whether your values are experimentally appropriate for a particular gas or temperature.

Interactive Calculator: Specific Heat Capacity

Use this tool to find the specific heat capacity of a material when the heat transfer, sample mass, and temperature change are known. The equation assumes a consistent unit system and no phase change.

Worked Example: Calculating Gamma

Suppose an ideal gas has a molar heat capacity at constant pressure of \(29.12\ \text{J/(mol·K)}\) and a molar heat capacity at constant volume of \(20.81\ \text{J/(mol·K)}\). Find the heat capacity ratio and compare the difference with R.

The calculated ratio is approximately 1.40. The difference, about \(8.31\ \text{J/(mol·K)}\), is close to the universal gas constant \(8.314\ \text{J/(mol·K)}\), allowing for rounding.

Applications of Heat Capacity

• Thermodynamics: Heat capacity helps calculate energy changes during heating, cooling, compression, and expansion.
• Engineering: Engineers use thermal properties when designing engines, heat exchangers, cooling systems, and thermal storage.
• Climate science: The high specific heat capacity of water helps oceans absorb and release large quantities of energy, moderating coastal and regional temperatures.
• Materials science: Heat capacity measurements help characterise materials and investigate changes in their structure and thermal behaviour.
• Chemistry laboratories: Calorimetry uses heat capacity to calculate energy transfer during chemical reactions and physical changes.
• Automotive systems: Coolants are selected partly for their thermal properties, helping transfer energy away from hot engine components.
• Energy storage: Materials with suitable heat capacities can store thermal energy for later use in buildings and industrial systems.

Common Mistakes in Heat Capacity Calculations

• Mixing specific and molar quantities: Do not use mass-based heat capacity with a molar formula without converting units.
• Ignoring the process condition: Always distinguish Cp at constant pressure from Cv at constant volume.
• Using inconsistent units: Ensure the heat, mass or mole count, and heat capacity use compatible units.
• Applying Cp − Cv = R to every substance: Mayer’s relation applies to molar heat capacities of an ideal gas under the stated assumptions.
• Forgetting the sign of heat: Heat absorbed by a system is positive under the convention used here; heat released is negative.
• Assuming heat capacities never change: Heat capacity can vary with temperature, particularly over wide temperature ranges.
• Confusing heat capacity with temperature: Heat capacity measures energy required for a temperature change; it is not a measure of how hot a substance is.

Summary of Important Heat Capacity Equations

Frequently Asked Questions About Heat Capacity

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