Cp and Cv for ideal gases and their relations

Specific heat

The specific heat capacity of a substance is the heat capacity of a sample of the substance divided by the mass of the sample. This implies that it is the amount of energy that must be added, in the form of heat, to one unit of mass of the substance in order to cause an increase of one unit in temperature. The specific heat capacity of gas, maybe significantly higher when it is allowed to expand as it is heated (specific heat at constant pressure) than when is heated in a closed vessel that prevents expansion (specific heat at constant volume). These two values are usually denoted by Cp and Cv, their quotient y = Cp/Cv is the heat capacity ratio. The heat capacity at constant pressure Cp is greater than the heat capacity at constant volume Cv, because when heat is added at constant pressure, the substance expands and works.

?Cp and Cv relations

?First relation

Cp/Cv = ?H/?U

Summary:

Out of the energy added to gas-only a portion of the energy which is the translational KE is only available to do work. The other portion of energy gets locked up in rotational and vibrational motions of molecules that have no ability to do work. Therefore Cp/Cv is a good indicator for actual energy available to do mechanical work.???

Specific heat constant pressure: Cp

Cp is the amount of heat energy released or absorbed by the unit mass of the substance with the change in temperature at constant pressure. In another word, it is the heat energy transfer at a constant pressure between system and surroundings.

Specific heat constant volume: Cv

During the small change in the temperature of a substance, Cv is the amount of heat energy released or absorbed by the unit mass of the substance with the change in temperature at constant volume. In another word, it is the heat energy transfer between a system and surrounding when there is no change in volume.

When heat QV is added to gas at constant volume, applying 1st law of thermodynamics ?Q = ?U+ ?W, where ?Q is the heat absorbed by the system, ?U is the change in internal energy of the system, and ?W is the work done by the system.

we have

QV = Cv ?T = ?U + W = ?U because no work is done.

Therefore, ?U = Cv ?T and CV = ?U/?T,?or, Cv ?T = ?U------------------ [1]

When heat QP is added at constant pressure,

we have

QP = Cp ?T = ?U + W = ?U + P ?V.

For infinitesimal changes this becomes

Cp ?T = ?U + P ?V, therefore since ?H = ?U + P ?V,

Cp ?T = ?H --------------------- [2]

Since from equation [1[ Cv ?T = ?U

Cp/Cv = ?H/?U ---------------------------------- [3]

Significance of Cp/Cv

Cp/Cv is an indicator of how much gas in adiabatic conditions with dQ=0 can extract heat internally to do work. Cp/Cv is an indicator of how much gas in adiabatic conditions with dQ=0 can extract heat internally to do work.

This can be further explained like this.

Total energy:?The total energy of a system = KE + PE

Kinetic energy [KL] is the movement of an object – or the composite motion of the components of an object and potential energy reflects the potential of an object to have motion.

What generates kinetic energy in a molecule? ?When bound in a stable state in an atom, an electron behaves mostly like an oscillating three-dimensional wave. These waves generate vibrations and kinetic energy in a molecule. Therefore, kinetic energy, a form of energy that an object or a particle has by reason of its motion. If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy. Kinetic energy is a property of a moving object or particle and depends not only on its motion but also on its mass. KE = ? mv2 [ m is mass and v is velocity]. The kind of motion may be a translation (or motion along a path from one place to another), rotation about an axis, vibration, or any combination of motions.

According to the theorem of Equipartition of energy, when heat is applied to a gas the added kinetic energy in the internal energy of the gas is equally distributed in all modes with each mode having average energy = ? KbT [Kb is Boltzmann constant and T is temperature]

Total energy E total = E translational + E rotational + E vibrational

While only translational KE does work all other forms of KE consume energy.

Cp/Cv = ?H/?U, tells us how much of the available energy out of the total internal energy of a gas is available to do wok, Q= U+W [sign of W can be both + and -] internally in an adiabatic system with Q=0. As said, out of the energy added to gas-only a portion of the energy that is the translational KE is available to do work. The other portion of energy gets locked up in rotational and vibrational motions of molecules that have no ability to do work. Therefore Cp/Cv is a good indicator for actual energy available to do mechanical work.???

Second relation

CP - CV = n R

Summary: Cp-Cv only shows that Cp exceeds Cv by an amount equivalent to R. Cp-Cv is constant regardless of the value of Cp and Cv. Specific heat is resistance to heat transfer. High specific heat is, therefore, an indicator of higher resistance to heat flow. Individually Cp and Cv mean a lot. Y [Gamma] = 1 + 2/[DOF] [ explained later] Higher the DOF the smaller the Cp/Cv = Gamma ratio. Therefore, the smaller the ratio of ?H/?U, the smaller the ability of the gas to extract energy to do work from the available internal energy

As explained, Cp is always more than Cv because when heat is added at constant pressure, the substance expands and works.

Let us go back to two fundamental equations

Cv ?T = ?U and CP ?T = ?H

We can expand ?H and write, [H=U+W]

?Cp ?T = CV ?T + P ?V

The value of ?U is substituted according to equation [1]

?From the ideal gas law, P V = n R T, we get for constant pressure (?P V) = P ?V + V ?P, we get

P ?V = n R ?T.

Substituting this in the equation Cp ?T = CV ?T + P ?V gives

Cp dT = CV dT + n R dT

Dividing dT out, we get

CP = CV + n R

This signifies as said above Cp always exceeds Cv by an amount n R [ n is moles of gas and R is the universal gas constant. But this does not say much externally unless probed further.

As the table shows while Cp-Cv for all gases is practically constant R=8.30 J/k/mol there is significant variation in the ratio of Cp/Cv. This can be explained like this.

Molecular motions: Concept of degrees of freedom

The internal energy of any gas is directly proportional to the temperature of the gas. It is the number of independent ways in which a molecule of gas can move in space in three axes, x,y, and z, this is technically the degrees of freedom [DOF] of a gas. As the number of atoms in a molecule increases their freedom to move in other modes like vibrational and rotational mode other than just a linear translational motion also increases. Without making it complex, monoatomic gases have DOF=3, diatomic gases have DOF=5 and triatomic gases have DOF =6

Each degree of freedom contributes 1/2kT per atom to the internal energy.

For monatomic ideal gases with N atoms, its total internal energy U is given as U=3/2NkT. For diatomic gases, U=5/2NkT, k is Boltzmann constant

Y [Gamma] = 1 + 2/[DOF]

Higher the DOF the smaller the Cp/Cv = Gamma ratio. Therefore, the smaller the ratio of ?H/?U, the smaller the ability of the gas to extract energy to do work from the available internal energy ??

Cp/Cv ratio for monoatomic, diatomic, triatomic is 1.67,1.4,1.33 respectively

Therefore, Cp-Cv only shows that Cp exceeds Cv by an amount equivalent to R. But if individually Cp and Cv are probed lots of information emerges. ?

Credit: Google

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