Heat Capacity: Definition, Equation, Relationship between CP and CV

Edited By Shivani Poonia | Updated on Jul 02, 2025 08:05 PM IST

Thermodynamics is the area of physics dealing with heat and other types of energy. In particular, thermodynamics explains how thermal energy is transformed into and from other kinds of energy, and its interrelation with matter. Any thermodynamic process is a process of energy transfer within a system or between systems. The properties of a thermodynamic process are pressure, temperature, and volume.

This Story also Contains

Constant Volume Process
Constant Pressure Process
Some Solved Examples
Summary

Heat Capacity: Definition, Equation, Relationship between CP and CV

Constant Volume Process

Representation on P-V diagram:

This process is represented on the P-V diagram by a vertical straight line as shown in the figure, since V₁=V₂.

Work done during the process:

$
\mathrm{W}=\int_{\mathrm{V}_1}^{\mathrm{V}_2} \mathrm{Pdv}
$

But $\mathrm{d} v=0$ for an isochoric process

$
\therefore \mathrm{W}=0
$

But $d v=0$ for an isochoric process

$\therefore \mathrm{W}=0$

Thus, work done during the constant volume process is zero which is also evident from the P-V diagram as no area is enclosed by the vertical line on the P-V diagram.

Constant Pressure Process

Representation on P-V diagram:

During this process the pressure or the gas remains constant therefore it is represented by a horizontal line on the P-V diagram. See figure.

Work done during the process:

$
\mathrm{W}=\int_{\mathrm{V}_1}^{\mathrm{V}_2} \operatorname{Pdv}
$

But $P$ is constant.
$\therefore$ Work done $=P \int_{V_1}^{V_2} d v=P\left(V_2-V_1\right)$

A rectangle on the P-V diagram represents the work done by the gas during the constant pressure process.

Some Solved Examples

Example 1: One mole of an ideal gas expands from state X to Y by three paths 1, 2, and 3 as shown in the figure below. If W₁, W_2, and W₃ are respective work done by ideal gas along the three paths then:

1)$W_1=W_2=W_3$

2)$W_3>W_1>W_2$

3) $W_3>W_2>W_1$

4)$W_1>W_2>W_3$

Solution

Work is a path function and not a state function. The area under the P-V curve gives Work done. As far as the magnitude is concerned, work done by gas will be maximum in path 3 because the area under the curve is highest in the case of path 3. The second highest area is under path 2 and the least area is of curve 1.
Hence, Option number (3) is correct

Example 2: The magnitude of work done by a gas that undergoes a reversible expansion along the path ABC shown in the figure is

1) 489Correct)

2)85

3)58

4)54

Solution

$\begin{aligned} & \mathrm{W}=(8-2)^2+\frac{1}{2}((12-8) \times(8-2)) \\ & \mathrm{W}=48\end{aligned}$

Example 3: What is the relation between the temperatures in the below graph which represents an isothermal expansion of gas at different temperatures?

1)$T 1>T 2>T 3$

2)$T 2>T 3>T 1$

3) $T 3>T 2>T 1$

4)No relationship can be established

Solution

In the isothermal process, Temperature is constant.
Also, we know for an ideal gas, $P V=n R T$

As we can see T is constant, we can say $P V=k$ (here k is constant).

This equation of $P V=k$ represents the equation of a hyperbola.

The higher the value of $k$ the farther the curve is from the origin. So we can conclude that
$T_3>T_2>T_1$
Hence, Option number (3) is correct

Example 4: Find out the magnitude of work (in kJ) done by one mole of an ideal gas for expansion.

1)-1

2)2

3) 6

4)60

Solution

Work is a path function and not a state function and the area under P-V curve gives work.

So, work will be the total area of the Trapezium

$\begin{aligned} & \therefore \mid \text { work } \left\lvert\,=\frac{1}{2} \times(\text { sum of parallel sides }) \times(\text { distance between the parallel sides })\right. \\ & \therefore \mid \text { work } \left\lvert\,=\frac{1}{2} \times 6 \times 20=60 \times 10^{-5}\right. \text { bar }- \text { lit } \\ & \therefore \mid \text { work } \mid=60 \times 10^5 \text { bar }- \text { lit }=60 \times 10^5 \times 10^{-3} \text { bar }-\mathrm{m}^3=6000 \mathrm{~J}\end{aligned}$

Thus, the magnitude of work done is 6 kJ

Example 5: One mole of an ideal monoatomic gas is subjected to changes as shown in the graph. The magnitude of the work done (by the system or on the system) is_________ J (nearest integer )

1) 6

2)7

3)4

4)6.5

Solution

$\begin{aligned} & \mathrm{I} \rightarrow \mathrm{II} \rightarrow \text { Isobaric } \\ & \mathrm{II} \rightarrow \mathrm{III} \rightarrow \text { Isochoric } \\ & \mathrm{III} \rightarrow \mathrm{I} \rightarrow \text { Isothermal } \\ & \mathrm{W}_{\mathrm{I}-\mathrm{II}}=-1[40-20]=-20 \text { Lit atm } \\ & \mathrm{W}_{\mathrm{II}-\mathrm{III}}=0 \\ & \mathrm{~W}_{\text {IV-I }}=2.303 \mathrm{nRt} \log \frac{\mathrm{V}_2}{\mathrm{~V}_1}\end{aligned}$

$
\begin{aligned}
& =2.303 \mathrm{PV} \log \frac{\mathrm{V}_2}{\mathrm{~V}_1} \\
& =2.303(1 \times 20) \log 2 \\
& =2.303 \times 20 \times 0.3010=13.818
\end{aligned}
$

W total $=-20+13.818=(-6.182$ lit atm $)=6.182$ lit atm

Summary

A thermodynamic process can simply be said to be a transfer of energy either within or between systems. The properties of the system in consideration are considered significant in these processes. The values of such properties at any given time describe the thermodynamic state of the system. A very simple example of a thermodynamic process could be the heating of water in a kettle. The heat from the surroundings—the stove—is transferred and absorbed by the kettle's system, which raises the temperature of the water.

Frequently Asked Questions (FAQs)

1. How does pressure affect the heat capacity of gases?

Pressure significantly affects the heat capacity of gases. At constant pressure (Cp), gases have a higher heat capacity than at constant volume (Cv). This is because when a gas is heated at constant pressure, it can expand, doing work against its surroundings. This work requires additional energy, increasing the overall heat capacity. At constant volume, no work is done, resulting in a lower heat capacity.

2. What is the significance of the heat capacity ratio (γ) in thermodynamics?

The heat capacity ratio (γ), also known as the adiabatic index, is the ratio of heat capacity at constant pressure (Cp) to heat capacity at constant volume (Cv). It's significant in thermodynamics because it determines how a gas behaves during adiabatic processes (where no heat is exchanged with the surroundings). The heat capacity ratio is used in calculations involving gas expansion, compression, and in understanding the behavior of gases in engines and other thermodynamic systems.

3. What is heat capacity and why is it important in chemistry?

Heat capacity is the amount of heat energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It's important in chemistry because it helps us understand how different materials respond to heat, which is crucial in many chemical processes and reactions. Heat capacity influences how quickly substances heat up or cool down, affecting everything from cooking to industrial processes.

4. What's the difference between specific heat capacity and molar heat capacity?

Specific heat capacity is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. Molar heat capacity, on the other hand, is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius. The main difference is the reference unit (gram vs. mole), which affects the numerical value and units of the heat capacity.

5. How does the equation q = mcΔT relate to heat capacity?

The equation q = mcΔT is the heat capacity equation, where q is the heat energy transferred, m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature. This equation shows that the heat energy transferred is directly proportional to the mass, specific heat capacity, and temperature change of the substance. It's used to calculate how much heat is needed to change the temperature of a given mass of a substance.

6. Why is the molar heat capacity of an ideal monatomic gas 3/2R at constant volume?

The molar heat capacity of an ideal monatomic gas at constant volume (Cv) is 3/2R, where R is the gas constant. This value comes from the equipartition theorem in statistical mechanics. For a monatomic gas, energy can only be stored in three translational degrees of freedom (motion in x, y, and z directions). Each degree of freedom contributes 1/2R to the heat capacity, resulting in a total of 3/2R. This illustrates how the simplicity of monatomic gases leads to a straightforward heat capacity value.

7. How does the concept of degrees of freedom relate to heat capacity?

Degrees of freedom in thermodynamics refer to the independent ways a molecule can store energy. Each degree of freedom contributes to the heat capacity of a substance. For gases, there are translational (movement in space), rotational (spinning), and vibrational (internal vibrations) degrees of freedom. More complex molecules have more degrees of freedom and thus higher heat capacities. This concept explains why diatomic gases have higher heat capacities than monatomic gases, and why solids and liquids generally have higher heat capacities than gases.

8. Why does the heat capacity of most substances increase with temperature?

The heat capacity of most substances increases with temperature due to the excitation of additional energy storage modes. As temperature rises, more vibrational modes in solids become active, and in gases, higher energy rotational and vibrational states become populated. This means more energy can be stored in these additional modes, increasing the overall heat capacity. However, this increase is not linear and tends to level off at high temperatures when all modes are fully excited.

9. How does isotopic substitution affect heat capacity?

Isotopic substitution can affect heat capacity, particularly in gases and at low temperatures. Heavier isotopes generally lead to slightly lower heat capacities. This is because heavier atoms have lower vibrational frequencies, which means their vibrational modes are less easily excited at a given temperature. The effect is most noticeable in hydrogen isotopes (protium, deuterium, tritium) due to their large relative mass differences. In solids at room temperature, the effect is usually small enough to be negligible for most practical purposes.

10. What is the relationship between heat capacity and entropy?

Heat capacity and entropy are closely related thermodynamic properties. The heat capacity of a substance determines how much its entropy changes with temperature. Mathematically, the change in entropy (dS) is related to heat capacity (C) by the equation dS = C dT/T, where T is temperature. Substances with higher heat capacities generally experience larger increases in entropy for a given temperature change. This relationship is fundamental in understanding how energy and disorder are related in thermodynamic systems.

11. How does the heat capacity of water compare to other common substances?

Water has an unusually high heat capacity compared to most other common substances. This means it can absorb or release a large amount of heat with relatively small changes in temperature. This property of water is crucial for regulating temperature in living organisms and in the Earth's climate system. For comparison, the heat capacity of water is about twice that of most oils and four times that of air.

12. Why do metals generally have lower heat capacities than non-metals?

Metals generally have lower heat capacities than non-metals because of their atomic structure. In metals, the outer electrons are delocalized and can move freely throughout the metal lattice. This means that when heat energy is added, it's quickly distributed among these electrons, resulting in a faster temperature increase. Non-metals, with their more localized electrons, require more energy to increase their temperature by the same amount, resulting in higher heat capacities.

13. How does heat capacity relate to the concept of thermal mass in buildings?

Heat capacity is directly related to thermal mass in buildings. Materials with high heat capacity, like concrete or brick, have high thermal mass. They can absorb and store large amounts of heat energy, helping to stabilize indoor temperatures. During the day, these materials absorb heat, keeping the interior cooler. At night, they release this stored heat, helping to maintain warmth. This property is used in passive solar design to improve energy efficiency in buildings.

14. What is the relationship between heat capacity and phase changes?

Heat capacity and phase changes are closely related. During a phase change (like melting or boiling), a substance absorbs or releases a large amount of heat energy without changing temperature. This is because the energy is used to break or form intermolecular bonds rather than increase kinetic energy. The heat capacity of a substance can appear to be infinitely large during a phase change, as temperature remains constant despite the addition or removal of heat.

15. How does molecular structure influence heat capacity?

Molecular structure greatly influences heat capacity. Molecules with more atoms generally have higher heat capacities because they have more ways to store energy (more degrees of freedom). For example, polyatomic molecules can store energy not just in translational motion, but also in rotational and vibrational modes. Additionally, the strength and type of bonds within a molecule affect how easily it can absorb and store thermal energy, thus impacting its heat capacity.

16. How does Einstein's model of solids explain heat capacity at low temperatures?

Einstein's model of solids explains the behavior of heat capacity at low temperatures by treating atoms in a solid as quantum harmonic oscillators. At very low temperatures, not all vibrational modes are excited, leading to a decrease in heat capacity. This model predicts that heat capacity approaches zero as temperature approaches absolute zero, in agreement with the third law of thermodynamics. It successfully explains the deviation from the classical Dulong-Petit law, which incorrectly predicts constant heat capacity at all temperatures.

17. What is the Dulong-Petit law and when does it break down?

The Dulong-Petit law states that the molar heat capacity of a solid element is approximately 3R (where R is the gas constant), or about 25 J/mol·K. This law works well for many solids at room temperature but breaks down at low temperatures and for light elements like beryllium and carbon. The breakdown occurs because the law assumes classical behavior of atoms, ignoring quantum effects that become significant at low temperatures or for light atoms with high vibrational frequencies.

18. What is the significance of the Debye temperature in relation to heat capacity?

The Debye temperature is a characteristic temperature of a solid that relates to its maximum vibrational frequency. It's significant in understanding heat capacity because it marks the temperature above which all vibrational modes in a solid are fully excited. Below the Debye temperature, heat capacity varies with temperature (approximately as T³ at very low temperatures). Above it, heat capacity approaches the classical Dulong-Petit value of 3R per mole. The Debye model thus provides a more accurate description of heat capacity over a wide temperature range than simpler models.

19. How does the concept of phonons relate to the heat capacity of solids?

Phonons are quantized vibrations in a solid's crystal lattice, and they play a crucial role in determining a solid's heat capacity. At low temperatures, only low-energy (long-wavelength) phonons are excited, resulting in a heat capacity that varies as T³. As temperature increases, higher-energy phonons become excited, leading to an increase in heat capacity. The Debye model, which treats these phonons quantum mechanically, successfully explains the temperature dependence of heat capacity in solids, including the approach to the Dulong-Petit limit at high temperatures.

20. How does intermolecular force strength affect heat capacity?

The strength of intermolecular forces significantly affects heat capacity, especially in liquids and solids. Stronger intermolecular forces generally lead to higher heat capacities because more energy is required to increase molecular motion against these forces. For example, water has unusually strong hydrogen bonds, contributing to its high heat capacity. In contrast, substances with weaker intermolecular forces, like many organic liquids, tend to have lower heat capacities. This relationship is crucial in understanding thermal properties of materials.

21. Why do some substances, like water, have anomalous heat capacities?

Water's anomalously high heat capacity is due to its unique molecular structure and hydrogen bonding. When heat is added to water, much of the energy goes into breaking and reforming hydrogen bonds rather than increasing molecular kinetic energy. This allows water to absorb a large amount of heat with only a small temperature change. Similar anomalies can occur in other substances with strong intermolecular forces or complex molecular structures that provide additional ways to store thermal energy.

22. How does the presence of impurities affect the heat capacity of a substance?

Impurities can significantly affect the heat capacity of a substance, especially in solids. They can introduce defects in crystal structures, creating additional vibrational modes and thus increasing heat capacity. In some cases, impurities can also reduce heat capacity by limiting the movement of atoms or molecules. The effect depends on the nature and concentration of the impurities. This phenomenon is important in materials science, where controlling impurities can be a way to tailor the thermal properties of materials.

23. How does the concept of lattice vibrations contribute to the heat capacity of crystals?

Lattice vibrations, or phonons, are the primary contributors to the heat capacity of crystalline solids. As temperature increases, more phonons are excited, increasing the solid's ability to store thermal energy. At low temperatures, only long-wavelength phonons are excited, leading to a T³ dependence of heat capacity. As temperature rises, shorter-wavelength phonons become excited, eventually leading to the classical Dulong-Petit limit. This concept is crucial in understanding the temperature dependence of heat capacity in solids.

24. What is the significance of the Einstein temperature in solid-state physics?

The Einstein temperature is a characteristic temperature in Einstein's model of solids, representing the temperature at which the highest-frequency vibrational modes of atoms in a crystal become significantly excited. It's important because it marks the transition between quantum and classical behavior in a solid's heat capacity. Below the Einstein temperature, quantum effects dominate, and heat capacity increases rapidly with temperature. Above it, the heat capacity approaches the classical Dulong-Petit limit. Understanding this helps in analyzing the thermal properties of materials across different temperature ranges.

25. What is the relationship between heat capacity and phase transitions?

Heat capacity and phase transitions are closely linked. During a phase transition, such as melting or boiling, the heat capacity of a substance appears to become infinitely large. This is because energy is being absorbed to break intermolecular bonds rather than to increase temperature. On a graph of heat capacity versus

26. How does the heat capacity of a substance change during a chemical reaction?

The heat capacity of a system can change significantly during a chemical reaction because the products often have different molecular structures and compositions compared to the reactants. These changes can affect the number of degrees of freedom and the strength of intermolecular forces, both of which influence heat capacity. For example, if a reaction produces more complex molecules, the heat capacity of the system might increase. Conversely, if it breaks down complex molecules into simpler ones, the heat capacity might decrease.

27. Why is the heat capacity at constant pressure (Cp) always greater than the heat capacity at constant volume (Cv) for gases?

For gases, Cp is always greater than Cv because when a gas is heated at constant pressure, it expands, doing work against its surroundings. This work requires additional energy, which is accounted for in Cp. In contrast, when a gas is heated at constant volume, no work is done, and all the heat goes into increasing the internal energy of the gas. The difference between Cp and Cv is equal to the gas constant R for an ideal gas, reflecting the extra energy needed for expansion work.

28. What is the significance of the Cp/Cv ratio in the behavior of gases?

The ratio of Cp to Cv, often denoted as γ (gamma), is called the heat capacity ratio or adiabatic index. It's significant because it determines how a gas behaves during adiabatic processes (where no heat is exchanged with the surroundings). This ratio affects the speed of sound in the gas, the efficiency of heat engines, and the behavior of gases during rapid compression or expansion. For monatomic ideal gases, γ = 5/3, while for diatomic gases, it's typically around 7/5. The closer γ is to 1, the more degrees of freedom the gas molecules have.

29. How does the concept of equipartition of energy relate to heat capacity?

The equipartition theorem states that in thermal equilibrium, energy is shared equally among all accessible degrees of freedom of a system. Each degree of freedom contributes 1/2kT to the average energy (where k is Boltzmann's constant and T is temperature). This principle is fundamental in understanding heat capacities. For example, it explains why the molar heat capacity of an ideal monatomic gas is 3/2R at constant volume – there are three translational degrees of freedom, each contributing 1/2R to the heat capacity.

30. How does the heat capacity of a substance change near its critical point?

Near the critical point, the heat capacity of a substance increases dramatically and can even approach infinity at the critical point itself. This phenomenon occurs because the distinction between liquid and gas phases becomes blurred, and small changes in temperature or pressure can cause large changes in the substance's state. The large heat capacity reflects the system's high sensitivity to energy input, as a small amount of added heat can cause significant changes in the substance's properties without much change in temperature.

31. What is the relationship between heat capacity and thermal conductivity?

Heat capacity and thermal conductivity are related but distinct properties. While heat capacity measures how much heat a substance can store, thermal conductivity measures how well it can transfer heat. Generally, materials with high heat capacity tend to have lower thermal conductivity, as they retain heat rather than quickly transmitting it. However, this is not always the case. For instance, water has both high heat capacity and relatively high thermal conductivity. Understanding both properties is crucial in applications like heat management in electronics or building insulation.

32. Why do gases have lower heat capacities compared to liquids and solids?

Gases generally have lower heat capacities than liquids and solids because they have fewer ways to store energy. In gases, energy is primarily stored in translational motion and, for polyatomic gases, rotational motion. Liquids and solids, however, can store energy in vibrational modes and intermolecular interactions in addition to these motions. The more complex structure of liquids and solids provides more degrees of freedom for energy storage, resulting in higher heat capacities.

33. How does anharmonicity in atomic vibrations affect heat capacity?

Anharmonicity in atomic vibrations refers to deviations from perfect harmonic oscillator behavior in solids. It becomes significant at high temperatures and leads to thermal expansion and an increase in heat capacity beyond what simple harmonic models predict. Anharmonic effects cause the interatomic potential to deviate from a perfect parabola, allowing atoms to vibrate with larger amplitudes as temperature increases. This results in a slight increase in heat capacity at high temperatures, beyond the Dulong-Petit limit, and is crucial for accurately modeling the thermal behavior of materials at extreme temperatures.