Question

The adiabatic bulk modulus of a diatomic gas at atmospheric pressure is
(A) $0N{m^{ - 2}}$
(B) $1N{m^{ - 2}}$
(C)$1.4 \times {10^4}N{m^{ - 2}}$
(D) $1.4 \times {10^5}N{m^{ - 2}}$

Answer 1

Hint: Bulk modulus = pressure/fractional volume=\[ - \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}\]
For a adiabatic process,
Bulk modulus = $ - \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}} = \gamma P$
Where,
$\Delta P$= Difference in pressure in \[Pa\]or in \[\dfrac{N}{{{m^2}}}\]
$\Delta V$=Difference in volume
$\gamma $= ratio of specific heats
The negative sign in the formula ensures that bulk modulus remains always positive.

Complete step by step answer:
We have to find the adiabatic bulk modulus of a diatomic gas at atmospheric pressure.
Now, we know that,
At normal temperature and pressure
\[NTP = 1.013Pa\]
\[ = {\text{ }}1.013 \times {10^5}\dfrac{N}{{{m^2}}}\]
\[\;\gamma = {\text{ }}1.4\]
Let us consider the bulk modulus. The bulk modulus is calculated as the pressure divided by the fractional volume.
We know that the formula for the bulk modulus for adiabatic process as,
 $K = - \dfrac{P}{{\dfrac{{\Delta V}}{V}}}$
$ = \gamma P$ ------------- (1)
 Let us now substitute the values in equation (1) we get,
$ \Rightarrow \left( {1.013 \times {{10}^5}\dfrac{N}{{{m^2}}}} \right)(1.4)$
Let us simplify the above expression.
$K = 1.4182 \times {10^5}\dfrac{N}{{{m^{ - 2}}}}$
Therefore, bulk modulus of diatomic gas at atmospheric pressure is \[1.4182 \times {10^5}N{m^2}.\]
Hence, option (D) is the correct option.
Additional information:
For isothermal gas, the bulk modulus is the ratio of change in pressure to that of the change in volume at a constant temperature T. For isothermal process=0
Thus, bulk modulus is given as,
\[{B_{isothermal}} = - \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}} = P\] , For ideal gas the bulk modulus is equal to the pressure.
•The adiabatic bulk modulus of an ideal gas is the ratio of change in pressure (\[\Delta P\] ) to that of the change in the volume (\[\Delta V\]) when changes are carried through an adiabatic process. Thus bulk modulus is given as,\[{B_{adiabatic}} = - \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}} = \gamma P\]
Where,
\[\gamma \]=ratio of specific heats.


Note:
•Bulk modulus is defined as a ratio of the increase in pressure to the resulting relative decrease in the volume.
•The bulk modulus is a thermodynamic quantity, and to specify a bulk modulus it is necessary to specify how the pressure varies during compression for example temperature kept constant, entropy kept constant, like this many other variations are possible.
•For adiabatic process bulk modulus is $\gamma $ (ratio of heat capacity) time’s pressure ($P$). Similarly, for an isothermal process bulk modulus is $P$(pressure)
•When gas is not an ideal gas then these formulas only give and approximation of bulk modulus.${B_{adiabatic}} = \gamma {B_{isothermal}} > {B_{isothermal}}$
The bulk modulus for the isobaric process is zero.