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calculate bulk modulus for ideal gas in isothermal and adiabatic process? 3 marks
Answers
Answer:
The volume of a gas changes when pressure applied on it is varied. The bulk modulus of a gas is defined as the ratio of volumetric stress to the volumetric strain i.e,
B
=
−
Δ
p
(
Δ
V
/
V
)
where
Δ
p
is a change in pressure and
Δ
V
is change in volume. The negative sign indicates that the gas compresses (volume decreases) when applied pressure is increased. The unit of bulk modulus is Pascal.
Bulk Modulus of Gases
Compression of a gas due to increases in applied pressure. The bulk modulus of a gas is
B
=
−
V
d
p
/
d
V
.
The pressure on a gas can be changed through an isothermal process (constant temperature) or an adiabatic process (constant entropy). A gas has different bulk moduli for isothermal and adiabatic processes.
The isothermal bulk modulus of an ideal gas is the ratio of change in pressure (
Δ
p
) to the fractional change in volume (
Δ
V
/
V
) at constant gas temperature
T
. Differentiate the ideal gas equation
p
V
=
n
R
T
to get
p
Δ
V
+
V
Δ
p
=
0
i.e.,
Δ
V
/
V
=
−
Δ
p
/
p
. Note that
Δ
T
=
0
because the temperature is constant. Substitute
Δ
V
/
V
in the defining equation to get the isothermal bulk modulus
B
i
s
o
t
h
e
r
m
a
l
=
−
Δ
p
(
Δ
V
/
V
)
=
p
.
The isothermal bulk modulus of an ideal gas is equal to its pressure.
The adiabatic bulk modulus of an ideal gas is the ratio of change in pressure to the fractional change in volume when these changes are carried out through an adiabatic process (no heat exchange with surrounding). The equation of state of an ideal gas for an adiabatic process is
p
V
γ
=
c
o
n
s
t
, where
γ
is the ratio of specific heats. Differentiate
p
V
γ
=
c
o
n
s
t
to get
p
γ
V
γ
−
1
Δ
V
+
Δ
p
V
γ
=
0
i.e.,
Δ
V
/
V
=
−
Δ
p
/
γ
p
. Substitute in the defining equation to get the adiabatic bulk modulus
B
a
d
i
a
b
a
t
i
c
=
−
Δ
p
(
Δ
V
/
V
)
=
γ
p
.
Note that
B
a
d
i
a
b
a
t
i
c
=
γ
B
i
s
o
t
h
e
r
m
a
l
>
B
i
s
o
t
h
e
r
m
a
l
. The bulk modulus of a gas in an isobaric process is zero.