8/19/2019 Heat Capacity at Constant Volume

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1) Heat capacity or thermal capacity :

 

is a measurable physical quantity equal to the ratio of the heat added to (or removed from) anobject to the resulting temperature change.[1] The ! unit of heat capacity is joule per "elvin  and

the dimensional form is#$%T&$'&1. pecific heat is the amount of heat needed to raise the

temperature of a certain mass by 1 degree elsius.

eat capacity is an e*tensive property of matter+ meaning it is proportional to the si,e of thesystem. -hen e*pressing the same phenomenon as an intensive property+ the heat capacity is

divided by the amount of substance+ mass+ or volume+ so that the quantity is independent of the si,e

or e*tent of the sample. The molar heat capacity is the heat capacity per unit amount (! unit:mole)

of a pure substance and the specific heat capacity+ often simply called specific heat+ is the heatcapacity per unit mass of a material. ccasionally+ in engineering conte*ts+ the volumetric heat

capacity is used.

Temperature reflects the average randomi,ed "inetic energy of constituent particles of matter (e.g.atoms or molecules) relative to the centre of mass of the system+ /hile heat is the transfer of energy

across a system boundary into the body other than by /or" or matter transfer. Translation+ rotation+

and vibration of atoms represent the degrees of freedom of motion /hich classically contribute tothe heat capacity of gases+ /hile only vibrations are needed to describe the heat capacities of most

solids[$] + as sho/n by the0ulong2etit la/. ther+ more e*otic contributions can come frommagnetic[3] and electronic[4] degrees of freedom in solids+ but these rarely ma"e substantialcontributions.

5or quantum mechanical reasons+ at any given temperature+ some of these degrees of freedom may

 be unavailable+ or only partially available+ to store thermal energy. !n such cases+ the specific heatcapacity is a fraction of the ma*imum. 6s the temperature approaches absolute ,ero+ the specific

heat capacity of a system approaches ,ero+ due to loss of available degrees of freedom. 7uantum

theory can be used to quantitatively predict the specific heat capacity of simple systems.

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2) What is specific heat at constant volume, cv? 

To ans/er this question+ /e should start /ith a more basic question:

-hat is internal energy+ and ho/ do you measure it8 !nternal energy is the molecular

activity of a substance the vibrational+ translational+ and rotational "inetic energy of 

the molecules of the substance. 9ou can imagine that this /ould be difficult to

measure+ but fortunately this energy results in something that /e can easily measure:

a change in the temperature of the substance. This leads us to the ne*t step:

measuring the internal energy change of a substance. (ote that /e are more

interested in the change in internal energy+ not the absolute value. -e /ill find that

the change in internal energy is all /e really need.) To measure this change+ /e turn

to the 1st #a/ of Thermodynamics for a closed system: ;q & ;/ <

de < du = >(?@) = >(2@)

!Ave /ritten the differential form above+ and divided through by the mass of the

system since it is closed+ and therefore the mass doesnAt change. o/+ /e are

interested in measuring du + the change in specific internal energy+ so letAs set up a

system that eliminates boundary /or"+ as /ell as changes in "inetic and potentialenergy. -eAll choose a rigid container that does not move during the process .

;q < du

!n the system /eAve chosen above+ the change in energy is equal to the heat /e add. -hat do youthin" /ill happen to the system /hen heat is added8 9ouAd e*pect the temperature to rise. !n fact+ it

turns out that for small amounts of heat added+ the temperature change is proportional to the

amount of heat added:

 

;q(< du)∝ dT

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To ma"e this an equality+ /e can add a proportionality constant to the relationship above:

 ;q(< du) < cvdT

so

cv < (du B dT)

-e call the proportionality constant cv+ the Cspecific heat at constant volume+D and hasdimensions of @nergyB(massE>T). 9ou may have heard of the term heat capacityF this is just the

specific heat times the mass.

What does cv mean, physically? 

The specific heat of a substance physically represents ho/ much the energy of a

substance ta"es in or gives off /hen its temperature changes. 5or e*ample+ if you heat ten

"ilograms of /ater from $G H to 1II H+ it ta"es more energy than to raise the temperature of anequal mass of air. This implies that the specific heat of /ater is larger than that of air. -e call

(lo/erJcase) c the specific heat because it is the heat capacity of a substance divided by its mass+

and all CspecificD properties are properties per unit mass+ or massJspecific properties. -hy call itconstantJvolume8 Kecause the specific heat is defined /hen the system /as held at constant

volume.

The heat capacity at constant volume of nR = 1 J·K −1 of any gas.

This represents the dimensionless heat capacity at constant volume; it is generally a function of

temperature due to intermolecular forces. For moderate temperatures, the constant for a

monoatomic gas is cv=!" #hile for a diatomic gas it is cv=$!" %see &. 'acroscopic measurements

on heat capacity provide information on the microscopic structure of the molecules.

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Molecular internal vibrations

When a gas is heated, translational kientic energy of molecules in the gas will increase. In addition, molecules in the gas may pick up many characteristic internal vibrations. Potential energy stored in these internal degrees of freedom contributes to specific heatof the gas.

 References :

1) wikipedia .

2) calpoly .

3)  Boundless Physics.