What is specific heat at constant volume?

Short Answer

Specific heat at constant volume is the amount of heat required to raise the temperature of one unit mass of a substance by one degree Celsius (or one Kelvin) while keeping the volume constant. It is represented by Cv.

When the volume is fixed, the substance cannot expand, so all the heat supplied is used only to increase its internal energy. Because no external work is done, the heat required at constant volume is less than the heat required at constant pressure.

Detailed Explanation :

Specific Heat at Constant Volume

Specific heat at constant volume, commonly written as Cv, is a fundamental thermodynamic property that describes how much heat energy must be supplied to a substance to increase its temperature by one degree when its volume is kept constant. This means the substance is not allowed to expand while being heated. The entire amount of heat supplied goes directly into increasing the internal energy of the substance.

In gases, this concept becomes especially important because gases expand noticeably when heated. But if the volume is fixed—for example, in a rigid container—the gas cannot expand, and therefore it cannot do any work against external pressure. This makes Cv smaller than Cp (specific heat at constant pressure), where the gas is free to expand and must do work on the surroundings.

Meaning of Heating at Constant Volume

Heating at constant volume means that no change in volume occurs during heating. This can happen only if the gas is kept inside a rigid container or chamber. Since the gas is not allowed to expand, it cannot push the surroundings, and therefore it cannot do any external work.

Mathematically, external work done by a gas is:

W = PΔV

At constant volume:

ΔV = 0 → W = 0

Because the gas does not do work, all the heat supplied increases the internal energy of the gas. This is why heating at constant volume is the simplest situation for studying changes in internal energy.

Mathematical Expression of Cv

Specific heat at constant volume is defined as:

Cv = (Heat supplied at constant volume) / (Mass × Rise in temperature)

For a gas of mass m heated at constant volume:

Qv = mCvΔT

Where:

  • Qv = heat supplied
  • Cv = specific heat at constant volume
  • ΔT = change in temperature

This tells us how much heat is needed to raise the temperature when the volume cannot change.

Why Cv Is Less Than Cp

Cv is always less than Cp for gases. The reason is:

  • At constant volume, no external work is done.
  • At constant pressure, the gas expands, and some heat must be used to do this work.

Therefore, heating a gas at constant pressure requires more heat than at constant volume. This leads to the famous thermodynamic relation known as Mayer’s relation:

Cp – Cv = R

Where R is the universal gas constant.

This shows that the difference between Cp and Cv is constant for ideal gases.

Internal Energy and Cv

The first law of thermodynamics states:

Q = ΔU + W

At constant volume:

W = 0 → Qv = ΔU

This means:

mCvΔT = ΔU

So Cv directly measures how the internal energy of the gas changes with temperature. For ideal gases, internal energy depends only on temperature, so Cv plays a key role in predicting energy changes.

Importance of Specific Heat at Constant Volume

Cv is an essential quantity in physics and engineering. It helps in many applications such as:

  1. Thermodynamic Calculations

Cv helps determine internal energy changes in:

  • Isothermal processes
  • Adiabatic processes
  • Isobaric and isochoric processes
  1. Gas Laws and Heat Transfer

Cv appears in:

  • Mayer’s relation
  • Adiabatic equation PVᵞ = constant
  • Calculations involving internal energy and heat flow
  1. Engine Cycles

In petrol (Otto) and diesel engines, certain steps occur at constant volume. Cv helps calculate heat supplied or rejected in these stages.

  1. Understanding Gas Behaviour

Cv provides an understanding of how much energy is stored inside gases when heated.

  1. Temperature Rise of Gases

If the volume is fixed, Cv helps calculate how quickly temperature rises when heat is added.

Cv for Ideal vs Real Gases

For ideal gases, Cv depends only on the type of gas and its molecular structure (monoatomic, diatomic, or polyatomic).
For example:

  • Monoatomic gases like helium have lower Cv.
  • Polyatomic gases have higher Cv because they have more ways to store internal energy.

For real gases, Cv varies slightly depending on temperature and pressure, because real gases do not follow the ideal gas assumptions perfectly.

Cv in Solids and Liquids

For solids and liquids, volume does not change much even when heated. Therefore, Cp and Cv are almost equal.
But for gases, the difference is significant because gases expand easily.

Conclus

specific heat at constant volume, Cv, internal energy, gas, thermodynamics

ion

Specific heat at constant volume (Cv) is the amount of heat needed to raise the temperature of a unit mass of a substance by one degree when its volume remains fixed. Because no work is done during this process, Cv always represents the heat fully used to increase internal energy. It is always smaller than Cp, and it plays a central role in thermodynamics, especially in understanding gas behaviour, internal energy changes, and engine cycles.