Two questions: Why are Cp and Cv state functions and why does Cp-Cv = R apply to only ideal gases?

There is a constraint to the Q/dT definition of specific heat. Q/dT cannot be integrated between two points because Q is path dependant and its path determines how much work Q can do. Thermodynamics stepped in to save the day, changing the definition of specific heat to include two distinct routes, one at constant pressure and the other at constant volume. The definition of specific heat was changed from nC = Q/dT to Cv = [U/T] v and Cp = [H/T] p by thermodynamics. Q, which is a path function, was replaced by H and U, which are state functions, in thermodynamics.

Summary

These are two questions that frequently cross our minds.

The fundamental problem with the definition of specific heat is simply, nC = ?Q/dT where C is the specific heat. But ?Q is an inexact differential path function. Q and W can follow any undefined path between two points. To enable Q to be a path function the path must be specified. Therefore, there are two paths made for heat flow one at constant pressure and the other at constant volume. That's why we have two specific heat Cp and Cv for gases. Why [Cp-Cv] = R applies for only ideal gases is simply because this equation is derived from the ideal gas equation PV = n RT.

Specific heat

A substance can't raise its temperature unless its molecules' energy demands are met. In other words, before the substance reaches internal thermal equilibrium, the energy supply to molecules is pre-taxed at the source. Only when a substance is in a state of thermal balance on the inside can it raise its temperature. Every substance has different types of molecules, different numbers of molecules, and different arrangements of molecules; as a result, each substance has its own demand for a specific amount of heat [energy] to achieve internal thermal equilibrium; as a result, each substance provides heat transfer resistance until its energy demand is met before increasing temperature. The specific heat is the resistance to heat transfer. The purpose of defining heat capacity is to connect changes in internal energy to observed changes in the variables that define system states. Because the only form of work a system made up of a single pure component can accomplish is atmospheric work, the first law is reduced to H = U - PV.

The heat capacity of a sample of a substance divided by the mass of the sample is the specific heat capacity (symbol cp). This means that it is the quantity of energy that must be given to one unit of mass of a substance in the form of heat to create a one-unit increase in temperature. Specific heat capacity is a term that is frequently used.

The specific heat capacity often varies with temperature and is different for each state of matter. The specific heat of a substance, especially a gas, may be 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 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 work.

Why Cp and Cv are state functions?

The fundamental problem with the definition of specific heat is IT it is simply, nC = dQ/dT where C is the specific heat. But dQ is an inexact differential path function. To enable Q to be a path function the path must be specified. Therefore, there are two paths made for heat flow one at constant pressure and the other at constant volume.

How do thermodynamics define Cp and Cv?

Thermodynamics concluded that Q is dependent on the process path, and that when work W is performed, things change. Thermodynamics intended C to remain a physical attribute of the material being processed, independent of the process path or whether work is being done. This is solved in thermodynamics by slightly altering the meaning of C. In thermodynamics, we correlate C with factors relevant to the state of the material being processed, such as specific internal energy U and specific enthalpy H, rather than path-dependent heat Q. The derivative of the specific internal energy U with respect to temperature at constant volume is used to establish the specific heat capacity at constant volume Cv:

Cv = [ ?U/?T] v

Thermodynamics defines specific heat capacity at constant pressure Cp as the derivative of the specific enthalpy H with respect to temperature at constant pressure:

Thermodynamics define Cp as

Cp = [ ?H/?T] p

Thus, thermodynamics made both Cp and Cv state functions. To summarize, Thermodynamics modified the definition of specific heat from nC = ?Q/Dt, to Cv = [ ?U/?T] v and Cp = [ ?H/?T] p. Thermodynamics replaced ?Q that is a path function by ?H and ?U which are state functions.

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Derivation of CP-Cv = R

What does this equation mean?

The equation means that when you supply energy H to a gas the energy goes to increase the internal energy U and for doing work. The difference between Cp and Cv always remains constant which is the universal gas constant

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

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

Credit: Google