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By the end of this section, you will be able to:
  • Solve problems involving heat transfer to and from ideal monatomic gases whose volumes are held constant
  • Solve similar problems for non-monatomic ideal gases based on the number of degrees of freedom of a molecule
  • Estimate the heat capacities of metals using a model based on degrees of freedom

In the chapter on temperature and heat, we defined the specific heat capacity with the equation Q = m c Δ T , or c = ( 1 / m ) Q / Δ T . However, the properties of an ideal gas depend directly on the number of moles in a sample, so here we define specific heat capacity in terms of the number of moles, not the mass. Furthermore, when talking about solids and liquids, we ignored any changes in volume and pressure with changes in temperature—a good approximation for solids and liquids, but for gases, we have to make some condition on volume or pressure changes. Here, we focus on the heat capacity with the volume held constant. We can calculate it for an ideal gas.

Heat capacity of an ideal monatomic gas at constant volume

We define the molar heat capacity at constant volume C V as

C V = 1 n Q Δ T , with V held constant .

This is often expressed in the form

Q = n C V Δ T .

If the volume does not change, there is no overall displacement, so no work is done, and the only change in internal energy is due to the heat flow Δ E int = Q . (This statement is discussed further in the next chapter.) We use the equation E int = 3 n R T / 2 to write Δ E int = 3 n R Δ T / 2 and substitute Δ E for Q to find Q = 3 n R Δ T / 2 , which gives the following simple result for an ideal monatomic gas:

C V = 3 2 R .

It is independent of temperature, which justifies our use of finite differences instead of a derivative. This formula agrees well with experimental results.

In the next chapter we discuss the molar specific heat at constant pressure C p , which is always greater than C V .

Calculating temperature

A sample of 0.125 kg of xenon is contained in a rigid metal cylinder, big enough that the xenon can be modeled as an ideal gas, at a temperature of 20.0 °C . The cylinder is moved outside on a hot summer day. As the xenon comes into equilibrium by reaching the temperature of its surroundings, 180 J of heat are conducted to it through the cylinder walls. What is the equilibrium temperature? Ignore the expansion of the metal cylinder.

Solution

  1. Identify the knowns: We know the initial temperature T 1 is 20.0 °C , the heat Q is 180 J, and the mass m of the xenon is 0.125 kg.
  2. Identify the unknown. We need the final temperature, so we’ll need Δ T .
  3. Determine which equations are needed. Because xenon gas is monatomic, we can use Q = 3 n R Δ T / 2 . Then we need the number of moles, n = m / M .
  4. Substitute the known values into the equations and solve for the unknowns.
    The molar mass of xenon is 131.3 g, so we obtain
    n = 125 g 131.3 g/mol = 0.952 mol,

    Δ T = 2 Q 3 n R = 2 ( 180 J ) 3 ( 0.952 mol ) ( 8.31 J/mol · ° C ) = 15.2 ° C .

    Therefore, the final temperature is 35.2 °C . The problem could equally well be solved in kelvin; as a kelvin is the same size as a degree Celsius of temperature change, you would get Δ T = 15.2 K .

Significance

The heating of an ideal or almost ideal gas at constant volume is important in car engines and many other practical systems.

Got questions? Get instant answers now!

Questions & Answers

is this book for preparing IIT or neet?
jyotirmayee Reply
is it possible to increase the temperature of a gas without adding heat to it?
gurpreet Reply
I'm not sure about it, but I think it's possible. If you add some form of energy to the system, it's a possibility. Also, if you change the pression or the volume of the system, you'll increase the kinetic energy of the system, increasing the gas temperature. I don't know if I'm correct.
playdoh
For example, if you get a syringe and close the tip(sealing the air inside), and start pumping the plunger, you'll notice that it starts getting hot. Again, I'm not sure if I am correct.
playdoh
you are right for example an adiabatic process changes all variables without external energy to yield a temperature change. (Search Otto cycle)
Brad
how to draw a diagram of a triode
Ssempala Reply
whate is fckg diagrame?
Arzoodan
why do we use integration?
Monalisa Reply
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan
what is the Elasticty
Heeran Reply
it is the property of the by virtue of it regains it's original shape after the removal of applied force (deforming force).
Prema
property of the material
Prema
which type of cable is suitable for patrol station wiring
Eric Reply
what is calorimeter
odionyenma Reply
heat measuring device
Suvransu
What is mean electric potential
Biren Reply
Electric density formula
Biren Reply
Int E•dA
Vineet
what is relation betweeen potential energy of a system and work done in bringing last chaege from infinity
Shikhar Reply
Einstein and his general theory of relativity, a very nice concept,which revolutionize our modern world.
Antares Reply
what's the theory of relativity?
Piyali
it related to universe time period
jyotirmayee
it is a theory in which time is taken as relative,for diff. motion of objects.
Antares
no !!!
jyotirmayee
it is E=mc'square
jyotirmayee
what is black hole?
jyotirmayee
E=mc^2 is just a reln which sums up everything dude
Antares
what does e=mc^2 stand for?explain
Piyali
m not dude
jyotirmayee
e for speed of light
jyotirmayee
c for speed of light
jyotirmayee
m for mass of object
jyotirmayee
black hole is a super massive object in space gravitation pull of which s so strong dt evn light cannot escape. John wheeler 1st detected it, n concept of it ws given by Einstein himself from his general theory of relativity. Basically v cn si 3 types of it super massive, interstellar n intermediate
Antares
which topic now u studying?
jyotirmayee
e defined? sorry it,s wrong
jyotirmayee
how black holes r formed?
Piyali
e is not speed of light
jyotirmayee
black holes formed when the centre of very massive star collapsed itself
jyotirmayee
e is energy.
Antares
what do u mean by quantum mechanics?
jyotirmayee
in most of d cases black holes r formed by massive collapsed star or star system
Antares
sahoo u don't seem to understand relative physics, plz study that first.
Antares
Apollo is the name of a satellite !!!!!
jyotirmayee
quantum mechanics is d study of physics describing nature at d smallest level of energy of atoms n subatomic particles.
Antares
qm means explains about the microscopic particles..
KRANTHI
nope name of d sun god
Antares
why the mercury used in thermometers?
jyotirmayee
Ali brother ur exactly spelling is wrong,,,,,
jyotirmayee
why the colour of tube light white?
jyotirmayee
why the mouth became red colour ,,by the regular eating of leaf called ,,betel combining with areca?
jyotirmayee
quantum mechanics is the study of the photon the light particle
Agrim
all formula for calculate specific latent heat of any substance
Idowu Reply
hello
Nigar
real
Nigar
c'mon guys.. let's talk Physics.
Yoblaze
mass multiplied with latent heat of a substance
Yoblaze
you both?
jyotirmayee
in which class?
jyotirmayee
physics is the only subject which underestimated chemistry ,,bio
jyotirmayee
how capacitor is made in inst?
Piyali
what is dipole moment
Bilal Reply
it is the product of electric charges and distance between the two charges
Shikhar
mishra true thanks dia
Ssempala
product of separation of the poles, the rest shikhar got is right
Brad
brad is separation and distance ,are they different?
Ssempala
What is actually a dipole? I know charge separated by a certain distance.... but what does that really mean? what happens in a dipole? why are the charge of same magnitude?
Monalisa
Dipoles forming as a result of the unbalanced distribution of electrons in asymmetrical molecules
Heeran
What is dielectric
Ashis Reply
its a type of medium. generally poor conductors. but their conductivity can be changed
vedanth
you just have to add impurities
vedanth
Thanks
Ssempala
grt
Ssempala
a material which behave as conductor
Shikhar
insulating material, energy level for electron transfer is very high e.g used to increase a magnetic field in a capacitor
Brad
What is the difference between specific heat capacity and heat capacity? Give the equations
elly Reply
Practice Key Terms 2

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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