QUESTION At the same temperature, how does the internal energy of 1 mole of heli
ID: 1462460 • Letter: Q
Question
QUESTION At the same temperature, how does the internal energy of 1 mole of helium gas compare with the internal energy of 1 mole of argon gas? (Select all that apply.)
a)They are the same, partly because both have the same average kinetic energy per atom.
b)The helium has more internal energy, partly because the average speed of its atoms is greater.
c)The argon has more internal energy, partly because its atomic mass and therefore its average kinetic energy is greater.
d)They are the same, partly because the average speed of the argon atoms and of the helium atoms are equal.
e)Although the helium atoms have a smaller mass m, they move faster, and have more internal energy because ½ mv2 depends linearly on the mass but quadratically on the speed.
f)They are the same, partly because both contain the same number of atoms.
What kinds of energy contribute to the total internal energy of a monatomic ideal gas? Review in the example what variables determine the average kinetic energy per molecule for a monatomic ideal gas. How do these variables differ for argon and helium at the same temperature? How does the number of atoms in 1 mole of argon compare with the number in 1 mole of helium, and what then does this imply about how their internal energies compare?
PRACTICE IT
Use the worked example above to help you solve this problem. A cylinder contains 2.15 mol of helium gas at 18.0°C. Assume that the helium behaves like an ideal gas.
EXERCISEHINTS: GETTING STARTED | I'M STUCK!
(a) Find the change in the internal energy, U, of the gas.
J
(b) Find the change in the average kinetic energy per atom.
J
Explanation / Answer
1) Q
a) They are the same, partly because both have the same average kinetic energy per atom.
f) They are the same, partly because both contain the same number of atoms
2) Q
*if you wanna skip all the reading and you're simply looking for the answers (although im hoping thats not the case)* :
a) 7803 J
b) 6.0265809 x 10^-21 J
c) 23409 J
Not a problem, ok lets start off by familiarizing you with the formulas needed for this operation
U = 3/2nRT
K.E.avg = 3/2kT
rms = (2U/m)
U = total internal energy of system
K.E.avg = average kinetic energy per molecule
rms = root mean square speed
n = number of moles (1.80 in this case)
R = 8.31451 J/mol K (this is an ideal gas constant)
T = temperature (in kelvin)
k = 1.38066 x 10^-23 J/K (Boltzmann constant)
m = mass (in kg)
ok now i've given you everything necessary to solve this particular question and now take you through the steps:
a) in order to find the total internal energy first convert 18 degrees Celsius to kelvin by adding 273 to it so you'll get 291K after that pretty much plug everything into the very first equation
U = 3/2nRT
U = 3/2(2.15)(8.31451)(291) after this calculation you'll get a result of: 7803J
that's the total internal energy
b) now you have to find the avg. K.E. per molecule. In order to do this pretty much just plug everything into the 2nd equation so you get
K.E.avg = 3/2kT
K.E.avg = 3/2(1.38066 x 10^-23)(291) this will bring you a result of: 6.0265809 x 10^-21 J
which is the avg K.E. per molecule
now.. on to the last step (:
c) In order to solve this you have to do a little bit of extra work, before plugging in anything you have to find the mass of the He which is .0086kg this is done by taking the molar mass of He which is about 4g, multiplying it by 2.15and then dividing that result in order to get it into kg
after you've done that part you'll need to plug in the mass and the total internal energy, which is the answer you received from part "a", into the last equation:
rms = (2)(7803)/(.0086) this will give you an rms speed of 1347.08 m/s HOWEVER, this is
not your desired
result
you're looking for the amount of energy that will be needed to double the rms speed, in order to find that you simply reverse the process... multiply 1347.08 m/s by 2 in order to find "double the speed"
your answer should be about 2694.16m/s afterward you must plug this answer into the equation and in place of the total internal energy you'll have the variable U so:
2694.16 = (2)(U)/(.0086) now.. simply solve for "U" and your answer should be about
31211.54 J this is also NOT your answer
all you have to do now is subtract 7803J from this result and you'll find the answer to part "c" which is the amount of energy required to double the rms speed which is about:
23409J (this may vary a little if you rounded in different places)
3) Q
(a)
The internal energy of an ideal gas is given by:
U = nCvT
The molar heat capacity at constant volume for a monatomic ideal gas is [1]:
Cv = (3/2)R
Hence,
U = (3/2)nRT = (3/2)nR(T_final - T_initial)
= (3/2) 4mol 8.3145Jmol¹K¹ (2.20×10²K - 2.70×10²K)
= -2494.35 J
(b)
Kinetic theory states that the average kinetic energy of single gas particle in an ideal gas is given by:
K = (3/2)k_bT
Hence,
K = (3/2)k_bT
= (3/2) 1.38065×10²³ JK¹ (2.20×10²K - 2.70×10²K)
= - 1.0354875×10²¹ J