# 2.3 Heat capacity and equipartition of energy

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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 $Q=mc\text{Δ}T,$ or $c=\left(1\text{/}m\right)Q\text{/}\text{Δ}T$ . 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

${C}_{V}=\frac{1}{n}\phantom{\rule{0.2em}{0ex}}\frac{Q}{\text{Δ}T},\text{with}\phantom{\rule{0.2em}{0ex}}V\phantom{\rule{0.2em}{0ex}}\text{held constant}.$

This is often expressed in the form

$Q=n{C}_{V}\text{Δ}T.$

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 $\text{Δ}{E}_{\text{int}}=Q.$ (This statement is discussed further in the next chapter.) We use the equation ${E}_{\text{int}}=3nRT\text{/}2$ to write $\text{Δ}{E}_{\text{int}}=3nR\text{Δ}T\text{/}2$ and substitute $\text{Δ}E$ for Q to find $Q=3nR\text{Δ}T\text{/}2$ , which gives the following simple result for an ideal monatomic gas:

${C}_{V}=\frac{3}{2}R.$

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 $20.0\phantom{\rule{0.2em}{0ex}}\text{°C}$ . 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 $20.0\phantom{\rule{0.2em}{0ex}}\text{°C}$ , 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 $\text{Δ}T$ .
3. Determine which equations are needed. Because xenon gas is monatomic, we can use $Q=3nR\text{Δ}T\text{/}2.$ Then we need the number of moles, $n=m\text{/}M.$
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
$n=\frac{125\phantom{\rule{0.2em}{0ex}}\text{g}}{131.3\phantom{\rule{0.2em}{0ex}}\text{g/mol}}=0.952\phantom{\rule{0.2em}{0ex}}\text{mol,}$

$\text{Δ}T=\frac{2Q}{3nR}=\frac{2\left(180\phantom{\rule{0.2em}{0ex}}\text{J}\right)}{3\left(0.952\phantom{\rule{0.2em}{0ex}}\text{mol}\right)\left(8.31\phantom{\rule{0.2em}{0ex}}\text{J/mol}·\text{°}\text{C}\right)}=15.2\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}.$

Therefore, the final temperature is $35.2\phantom{\rule{0.2em}{0ex}}\text{°C}$ . 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 $\text{Δ}T=15.2\phantom{\rule{0.2em}{0ex}}\text{K}\text{.}$

## Significance

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

#### Questions & Answers

What is differential form of Gauss's law?
Rohit Reply
help me out on this question the permittivity of diamond is 1.46*10^-10.( a)what is the dielectric of diamond (b) what its susceptibility
OLUWA Reply
a body is projected vertically upward of 30kmp/h how long will it take to reach a point 0.5km bellow e point of projection
Abu Reply
i have to say. who cares. lol. why know that t all
Jeff
is this just a chat app about the openstax book?
Lord Reply
kya ye b.sc ka hai agar haa to konsa part
MPL Reply
what is charge quantization
Mayowa Reply
it means that the total charge of a body will always be the integral multiples of basic unit charge ( e ) q = ne n : no of electrons or protons e : basic unit charge 1e = 1.602×10^-19
Riya
is the time quantized ? how ?
Mehmet
What do you meanby the statement,"Is the time quantized"
Mayowa
Can you give an explanation.
Mayowa
there are some comment on the time -quantized..
Mehmet
time is integer of the planck time, discrete..
Mehmet
planck time is travel in planck lenght of light..
Mehmet
it's says that charges does not occur in continuous form rather they are integral multiple of the elementary charge of an electron.
Tamoghna
it is just like bohr's theory. Which was angular momentum of electron is intral multiple of h/2π
Aditya
determine absolute zero
OFERE Reply
The properties of a system during a reversible constant pressure non-flow process at P= 1.6bar, changes from constant volume of 0.3m³/kg at 20°C to a volume of 0.55m³/kg at 260°C. its constant pressure process is 3.205KJ/kg°C Determine: 1. Heat added, Work done, Change in Internal Energy and Change in Enthalpy
Opeyemi Reply
U can easily calculate work done by 2.303log(v2/v1)
Abhishek
Amount of heat added through q=ncv^delta t
Abhishek
Change in internal energy through q=Q-w
Abhishek
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
Gwam Reply
what is copper loss
timileyin Reply
this is the energy dissipated(usually in the form of heat energy) in conductors such as wires and coils due to the flow of current against the resistance of the material used in winding the coil.
Henry
it is the work done in moving a charge to a point from infinity against electric field
Ashok Reply
what is the weight of the earth in space
peterpaul Reply
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
Ifeoma Reply
every law is important, but first law is conservation of energy, this state is the basic in physics, in this case first law is more important than other laws..
Mehmet
First Law describes o energy is changed from one form to another but not destroyed, but that second Law talk about entropy of a system increasing gradually
Mayowa
first law describes not destroyer energy to changed the form, but second law describes the fluid drection that is entropy. in this case first law is more basic accorging to me...
Mehmet
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe

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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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