Question 6
(a)
The pressure p and volume V
of an ideal gas are related to the density ρ of the gas by the expression
p = 1/3 ρ〈c2〉.
(i)
State what is meant by the symbol 〈c2〉. [1]
(ii)
Use the expression to show that the mean kinetic energy EK of a gas molecule is given by
EK = 3/2 kT
where k
is the Boltzmann constant and T is
the thermodynamic temperature. [3]
(b)
(i) An ideal gas containing 1.0 mol of molecules is heated at constant
volume.
Use the expression in (a)(ii)
to show that the thermal energy required to raise the
temperature of the gas
by 1.0 K has a value of 3/2 R, where R
is the molar gas constant. [3]
(ii)
Nitrogen may be assumed to be an ideal gas. The molar mass of
nitrogen gas is 28 g mol-1. Use the answer in (b)(i)
to calculate a value for the specific heat capacity, in J kg-1 K-1, at constant volume for
nitrogen. [2]
[Total: 9]
Reference: Past Exam Paper – June 2017 Paper 42 Q2
Solution:
(a)
(i)
It is the
mean square speed.
(ii)
{Ideal
gas equation:}
pV = NkT or pV = nRT
{Density = mass / volume = Nm / V
Where N is the number of molecules and m is the mass of 1 molecule}
ρ = Nm / V
or
ρ = nNAm / V and k = nR / N
{Substituting
the formula for density in the expression in part (a) gives:
p =
1/3 Nm <c2> / V
So,
pV = 1/3 Nm <c2>
But
(also,) pV = NkT
So,
1/3
Nm <c2> = NkT
m <c2>
= 3kT
Multiply
by ½ ,
½ m
<c2> = 3/2 kT
EK
= 3/2 kT}
(b)
(i)
{ΔU = q + w
The volume is
constant, so there is no change in volume
External work done,
w = pΔV = 0
So, ΔU = q}
no (external) work done or ΔU = q or w = 0
{The
thermal energy q required is equal to the change in internal energy.
Internal
energy = kinetic energy + potential energy
For
an ideal gas, potential energy = 0
So,
internal energy = kinetic energy
KE
of one molecule = 3/2 kT
1 mole contains NA (Avogadro
constant) molecules.
KE
of 1 mole of molecules = NA ×
3/2 kT}
q = NA × (3 / 2)k × 1.0
{since
NA × k = R,}
NAk = R so, q = (3 / 2)R
(ii)
{s.h.c = heat capacity / mass
Heat capacity is the heat required to raise the
temperature by 1 K. So, this is the energy calculated above.
Mass of 1 mole of nitrogen = 0.028 kg}
specific heat capacity = {(3 / 2) × R} / 0.028
s.h.c.
= 450 J kg-1 K-1
Thank you.
ReplyDeleteHowever, I'd like to ask a question. Is "Heat Capacity" the amount of energy needed to raise the temperature of the whole gas sample by 1K? And not the energy needed to raise the temperature by 1K of 1 mole or 1 kg of the gas sample?
yes. heat capacity is for the while gas
Deletespecific heat capacity is for one kg.