Lesson 45 questions – Boltzmann Constant
(
Avogadro’s number (NA) = 6.02x1023 mol-1
Ideal gas constant (R) = 8.31 Jmol-1K-1
1.
/44)………%………
Use the data given on this page to calculate the value of the Boltzmann constant (k)
Boltzmann constant (k)
= R/NA
= 8.31/6.02x1023
k = ……1.38x1023 ……… units… JK-1………. (3)
2.
Calculate:
(a)
the kinetic energy of an individual gas molecule of mass 3.5x10-26 kg
moving at a speed of 600 ms-1.
2
½ mv = ½ x 3.5x10-26x6002
Kinetic energy = …6.3x10-21 ………….. J (3)
(b)
the kinetic energy of the gas molecules in a cylinder at a temperature of
20oC
3/2 kT = 3/2x1.38x10-23x293
Kinetic energy = ……6.07x10-21 ……….. J (3)
3.
A cylinder contains 2 moles of a gas composed of molecules with a mass of
-26
2.5x10 kg moving with an r.m.s. speed of 500 ms-1. Calculate the temperature of the gas.
RT
T
= 1/3 mNc2
= 1/3 [2xNAmc2/R]
= 1/3 x[2x6.02x1023x2.5x10-26x5002]
= 302 K
Temperature = …29……….. oC (4)
4.
A container of volume 1.5 m3 is full of gas with a density of 1.8 kgm-3 at a pressure
of 1.4x105 Pa.
(a)
Calculate the root mean square velocity of the molecules within the gas.
2
1.4x105x1.5x3/1.8 = <c>2
<c> = 592 ms-1
root mean square velocity = ……592………… ms-1 (3)
(b)
If the mass of an individual molecule of the gas is 3.0x10-27 kg calculate the
number of molecules in the container
Density (
N = V/m
= mN/V
= 1.8x1.5/3x10-27
Number of molecules = ……9x1026…………..(3)
5.
A deuterium plasma contains deuterium ions of mass 4.9x10-27 kg at a temperature
of 150x106 K.
Calculate:
(a) the r.m.s. speed and
RT = 1/3 mNc2
<c>2 = 3RT/mN
= [3x8.31x150x106]/[4.9x10-27x6.02x1023]
= 1.2677x1012
<c> = 1.13x106 ms-1
r.m.s. speed = ……1.13x106 ………….. ms-1 (3)
(b) the kinetic energy of a deuterium ion in the plasma
½ mv2 =½ (4.9x10-27)(1.13x106) 2
Kinetic energy = ……3.12 x10-15……………….. J (3)
6.
The mean kinetic energy of a gas molecule at an absolute temperature T is given by:
kinetic energy = 3RT/2NA
where R is the molar gas constant and NA is the Avogadro constant.
a)
Calculate the mean kinetic energy of gas atoms at 0 °C.
Kinetic energy = ………5.7x10-21…………….. J (2)
b)
Determine the speed of carbon dioxide molecules at 0 °C. The molar mass
of carbon dioxide is 44 g.
Speed = ……390………. ms-1 (5)
c)
Calculate the change in the internal energy of one mole of carbon dioxide
gas when its temperature increases from 0 °C to 100 °C.
change in the internal energy = ……1.2kJ………… (3)
SOME
7
from the kinetic theory, gas pressure is given by:
P = ⅓ρ<c>2
a)
State the ideal gas equation for n number of moles
PV = nRT
(1)
b)
By using the equation for pressure given by the kinetic theory, the ideal gas
equation and the definition of kinetic energy show that E = 3/2kT for the mean translational
kinetic energy of an atom.
P = ⅓ρ<c>2
2
Is the same as P  Nmv / 3V .
PV = nRT
(1)
Therefore:
nRT = 1/3 Nm<c>2
(1)
Now,
KE = 1/2m v2
(1)
Rearranging
3nRT / N = m<c>2
(1)
So,
1/2 m<c>2= (3/2) nRT/N
(1)
We know that
NA = N/n so n/N = 1/NA
(1)
(N is the number of molecules in a volume of gas, n is the amount of gas in moles,
NA is Avagadro’s number).
Therefore:
E = 1/2 m<c>2= (3/2) RT / NA = (3/2)kT
(1)
(7)
b)
What can we deduce about the relationship between the mean translational
kinetic energy of a molecule and the thermodynamic temperature of a gas?
……… the mean kinetic energy of a molecule of an ideal gas is proportional to the
thermodynamic temperature ………………………………………………………………...
…………………………………………………………………………………………… (1)