Understanding Specific Heat Capacity at Constant Pressure and Volume

By Kranchi Prajapati, Laboratory Technologist

C_p and C_v are reliable guides in the fascinating domain of thermodynamics and material science, navigating us through the unexplored realm of temperature and pressure. They uncover the mysteries of heat transfer and phase transitions, aiding our comprehension of it all. Let’s consider a simple scenario that captures the curiosity of a scientist during winter: Homes can be heated by circulating hot water through radiators. Imagine a young scientist pondering, “What mass of water would yield the same amount of heat when cooled from 90.0 to 20.0 °C as the heat released when 100 g of steam is cooled from 110 °C to 100 °C?” But, before delving into the solution, let’s grasp some fundamental theory.

What is Specific Heat Capacity?

Specific heat capacity, a fundamental thermodynamic property denoted as “c,” quantifies the energy required to elevate the temperature of a unit mass of a substance by a given temperature increment. It is typically expressed in units of Joules per kilogram per Kelvin (J/(kg·K)) in the International System of Units (SI) or in calories per gram per degree Celsius (Cal/(g·°C)) in some contexts. This property characterizes a material’s inherent capacity to absorb, store, and release thermal energy. It is pivotal in various scientific and engineering applications, particularly heat transfer analysis and thermodynamic calculations. Specific heat capacity is instrumental in comprehending how different substances respond to changes in temperature.

The contrast in temperature perception when walking on hot sand versus stepping into water under the same solar exposure can be attributed to the varying heat capacities of these materials. Water’s high heat capacity lets it absorb heat energy without significant temperature rise, providing a relatively cooler sensation. In contrast, sand’s lower heat capacity causes it to heat up quickly, resulting in a warmer sensation when walked upon.

To elucidate this concept, consider a comparative scenario involving two distinct containers: one filled with air and the other with water. When subjecting these containers to an identical heat source, a conspicuous dissimilarity emerges in their thermal response. In the case of dry air, it requires approximately 1.005 kJ of heat energy to raise the temperature of 1 kg by 1°C. Conversely, 1 kg of water mandates a substantially higher energy input, about 4.18 kJ, to effectuate the same temperature elevation. This discrepancy exemplifies the marked contrast in specific heat capacities between air and water, underscoring water’s considerably greater heat storage capacity. Such a contrast in specific heat capacities has profound implications for heat dynamics, influencing the rate and magnitude of temperature changes within different substances.

Introduction of C_p and C_v:

The specific heat of a substance can be determined through an isochoric (constant-volume) or an isobaric (constant-pressure) process. These specific heats are denoted as C_v and C_p, respectively, and they are inherent characteristics of a material. They play a pivotal role in computing alterations in a substance’s specific internal energy (U) and specific enthalpy (H) in processes encompassing ideal gases, liquids, and solids. These properties provide a framework for evaluating how energy content changes within a substance under different thermodynamic conditions, making them valuable tools for understanding and quantifying the thermal behavior of matter.

Importance of C_p and C_v:

The context and the objective of the experiment determine the heat capacity measurement. C_p is appropriate for experiments that take place at a constant pressure, such as atmospheric phenomena, combustion reactions, or chemical engineering operations. C_v is appropriate for experiments that take place at a constant volume, such as in closed containers, internal combustion engines, or thermodynamic cycles. Maintaining awareness regarding the chosen experimental conditions is crucial, specifically whether the apparatus is configured for constant pressure or volume measurements.
For instance, a chemical batch reactor operates on the principle of constant volume, whereas a flow reactor operates on the principle of constant pressure. The safety assessments for processes conducted under constant volume and pressure conditions necessitate a thorough analysis of the experimental framework’s heat, work, and internal energy contributions. This entails a meticulous comparison with the heat capacities inherent to the system. These heat capacities serve as pivotal parameters in computing the temperature variations within the system consequent to the absorption or release of a specified quantity of heat. The resultant temperature changes wield a substantial influence over the system’s pressure and volume, potentially engendering considerable safety hazards, such as explosions or leaks. Consequently, a comprehensive comprehension of the distinctions between constant volume and constant pressure processes is imperative, as it significantly informs the safety calculations integral to the experimental protocol.

Specific Heat Capacity at Constant Volume (C_v):

Constant-volume specific heat, denoted as C_v, represents the amount of energy needed to increase the temperature of a single unit mass (1 kg) of a material by one degree (1°C or 1 K) within an isochoric process. In mathematical terms, it is defined as:

Specific Heat Capacity at Constant Pressure (C_p):

The specific heat at constant pressure, denoted as C_p, signifies the energy necessary to increase the temperature of a material’s unit mass (1 kg) by one degree (1°C or 1 K) in an isobaric process. In a mathematical context, it is articulated as:

Ideal Gas and relationship between C_p and C_v:

The derivation concludes that the difference between C_p and C_v is the gas constant(R). When dealing with real gases, the interaction dynamics among gas molecules, particularly Van der Waals forces, introduce nuances that necessitate consideration. In such cases, the relationship between C_p and C_v deviates from ideal behavior, thereby warranting a more comprehensive examination of these heat capacities under real gas conditions.
The van der Waals equation is a modification of the ideal gas law that considers the intermolecular forces and finite size of gas molecules. It is given by:

To derive the relationship between C_P and C_v using the van der Waals equation, we need to calculate the partial derivatives of the internal energy U with respect to temperature T and volume V. This gives us:

For a van der Waals gas, the partial derivatives can be calculated using the equation of state and the expression for the internal energy. After some algebraic manipulation, we get:

This expression shows that C_P – C_V depends on the parameters a and V, which are specific to each gas. Liquids and solids are typically regarded as incompressible substances due to their minimal volume changes in response to alterations in pressure and temperature. In the case of liquids and solids, the disparity between the constant-volume specific heat (C_v) and the constant-pressure specific heat (C_p) is typically negligible; thus, both heat capacities are often treated as nearly equal.

Solution:

In the problem we need to determine the mass of water required to cool it down from 90°C to 20°C.

We know, C_p (water) = 4.01 J/g °C; C_p (Steam) = 2.01 J/g °C.

Now, we can assume that heat loss by the water when it cools down = heat gained by steam

7.16 g mass of water would yield the same amount of heat when cooled from 90.0 to 20.0 °C as the heat released when 100 g of steam is cooled from 110 °C to 100 °C

Real Life examples:

1. Automobile Engines: Engineers use specific heat capacities (Cp and Cv) to optimize internal combustion engine design, ensuring efficient energy conversion during fuel combustion.
2. Air Conditioning Systems: Changes in Cp and Cv are harnessed in air conditioning and refrigeration systems to regulate temperature by compressing and expanding gases.
3. Chemical Reactions: Specific heat capacities play a pivotal role in calculating the heat of chemical reactions (ΔH), facilitating efficient process design .
4. Space Exploration: Cp and Cv are crucial in designing rocket engines, enabling precise energy calculations for space missions.
5. Materials Science: Heat capacity is vital for understanding material responses to temperature variations, influencing advancements in metallurgy, ceramics, and composites.

At Thermal Analysis Labs (TAL), our expertise extends to comprehensively analyzing a broad spectrum of materials, encompassing intricate mixtures and elemental fabrics. For further insights and detailed information, we invite you to explore our services at Thermal Analysis Labs, a division of C-Therm Technologies Ltd., accessible via our official website: https://www.ctherm.com/thermal-analysis-labs.
Measurement of Cp and Cv using DSC:
Differential Scanning Calorimetry (DSC): DSC is a precise method for determining heat capacities, including Cp and Cv, for many substances. It involves heating or cooling a sample while measuring the heat flow, allowing for calculating heat capacities.

At Thermal Analysis Lab, we harness the remarkable capabilities of Rigaku DSC (Differential Scanning Calorimetry) to unlock the intricacies of specific heat capacities (Cp) and (Cv). Rigaku DSC is a powerful thermal analysis tool that measures a sample’s heat flow in and out as it undergoes temperature changes. This precise and versatile technology enables us to accurately analyze various materials, from polymers to pharmaceuticals.
The recognition of Rigaku DSC within the Thermal Analysis Lab is a testament to our commitment to cutting-edge analytical techniques. We take pride in offering this state-of-the-art service to various companies across industries. Whether you are in materials science, pharmaceuticals, or engineering, our Rigaku DSC testing services provide invaluable insights into your materials, allowing you to optimize product quality and innovation. Join the ranks of leading companies that trust Thermal Analysis Lab for their rigorous testing needs and experience the difference that advanced thermal analysis can make in your research and development endeavors.

Learn more about Rigaku DSC here, You can also watch a webinar on DSC explaining depth in calorimetry at, Also, looking to get more in depth knowledge of thermal conductivity visit our webinar website at Webinars – C-Therm Technologies Ltd. (ctherm.com).
To receive a quote for a testing contract, visit us here.

---

Reference

1. Chemistry, LibreTexts. 8.4.1: Practice Problems- Enthalpy and Hess’ Law. n.d. LibreTexts Chemistry. 2023. https://chem.libretexts.org/Courses/Oregon_Tech_PortlandMetro_Campus/OT_-_PDX_-_Metro%3A_General_Chemistry_I/08%3A_Thermochemistry/8.04%3A_Standard_Enthalpy_and_Hess_Law/8.4.01%3A_Practice_Problems-_Enthalpy_and_Hess_Law.
2. Moran, Michael J and Howard N. Shapiro. Funamentals of Engineering Thermodynamics. 5th . John Wiley & Sons, Inc., n.d.
3. Powers, Joseph M. “LECTURE NOTES ON THERMODYNAMICS.” 31 October 2023. nd.edu. Ed. Department of Aerospace and Mechanical Engineering. <https://www3.nd.edu/~powers/ame.20231/notes.pdf>.
4. Tatum, Jeremy. Cp Minus Cv. n.d. <https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Heat_and_Thermodynamics_%28Tatum%29/10%3A_The_Joule_and_Joule-Thomson_Experiments/10.04%3A_CP_Minus_CV>.
5. Yan, Claire Yu. Introduction to Engineering. Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License,, n.d. PDF. November 2023. <https://pressbooks.com>.