Relations between heat capacities - AbsoluteAstronomy.com

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, the heat capacity

Heat capacity

Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

at constant volume,

, and the heat capacity at constant pressure,

, are extensive properties that can be written in dimensions of energy/degree temperature.

Relations

The laws of thermodynamics

Laws of thermodynamics

The four laws of thermodynamics summarize its most important facts. They define fundamental physical quantities, such as temperature, energy, and entropy, in order to describe thermodynamic systems. They also describe the transfer of energy as heat and work in thermodynamic processes...

imply the following relations between these two heat capacities (Gaskell 2003:23):

Here

is the thermal expansion coefficient:

is the isothermal compressibility:

and

is the isentropic compressibility:

A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is:

where ρ is the density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of the substance under the applicable conditions.

The corresponding expression for the ratio of specific heat capacities

Heat capacity ratio

The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

remains the same since the thermodynamic system

Thermodynamic system

A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....

size-dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Thus:

The difference relation allows one to obtain the heat capacity for solids at constant volume which is not readily measured in terms of quantities that are more easily measured. The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio.

Derivation

If an infinitesimal small amount of heat

is supplied to a system in a quasistatic way then, according to the second law of thermodynamics

Second law of thermodynamics

The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

, the entropy change of the system is given by:

Since

where C is the heat capacity, it follows that:

The heat capacity depends on how the external variables of the system are changed when the heat is supplied. If the only external variable of the system is the volume, then we can write:

From this we see that:

Expressing dS in terms of dT and dP similarly as above leads to the expression:

We can find the above expression for

by expressing dV in terms of dP and dT in the above expression for dS. We have

which gives

and we see that:

The partial derivative

can be rewritten in terms of variables that do not involve the entropy using a suitable Maxwell relation. These relations follow from the fundamental thermodynamic relation:

It follows from this that the differential of the Helmholtz free energy

is:

This means that

and

The symmetry of second derivatives

Symmetry of second derivatives

In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...

of F with respect to T and V then implies

allowing us to write:

The r.h.s. contains a derivative at constant volume, which can be difficult to measure. We can rewrite it as follows. In general, we have:

Since the partial derivative

is just the ratio of dP and dT for dV = 0, we can obtain this by putting dV = 0 in the above equation and solving for this ratio. We then obtain:

which yields the expression:

The expression for the ratio of the heat capacities can be obtained as follows. We have:

The partial derivative in the numerator can be expressed as a ratio of partial derivatives of the pressure w.r.t. temperature and entropy. If in the relation

we put

and solve for the ratio

we obtain

. Doing so gives:

We can similarly rewrite the partial derivative

by expressing dV in terms of dS and dT, putting dV equal to zero and solving for the ratio

. If we substitute that expression in the heat capacity ratio expressed as the ratio of the partial derivatives of the entropy above, we obtain:

Taking together the two derivatives at constant S:

Taking together the two derivatives at constant T:

We can thus write

Ideal gas

We now try to obtain an expression for

for an ideal gas

Ideal gas

An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

.

An ideal gas

Ideal gas

has the equation of state

Equation of state

In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

where

P = pressure

V = volume

n = number of moles

R = universal gas constant

T = temperature

The ideal gas

Ideal gas

equation of state

Equation of state

can be easily arranged to give:

The following partial derivatives are easily obtained from the above equation of state

Equation of state

The following simple expressions are obtained for thermal expansion coefficient

and for isothermal compressibility

We now calculate

for ideal gases from the previously-obtained general formula:

Substituting from the ideal gas

Ideal gas

equation gives finally:

where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows:

This result would be consistent if the specific difference were derived directly from the general expression for

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