In the isothermal condition, the isothermal bulk modulus of an ideal gas is equal to ______.

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  1. constant
  2. pressure
  3. temperature
  4. viscosity

Answer (Detailed Solution Below)

Option 2 : pressure
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Detailed Solution

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Concept:

Compressibility is the reciprocal of the bulk modulus of elasticity.

Compressibility (p) = 1/K, and K = bulk modulus of Elasticity

\({\rm{K}} = \frac{{{\rm{Increase\;of\;pressure}}}}{{{\rm{Volumetric\;strain}}}} = \frac{{{\rm{dP}}}}{{\frac{{ - {\rm{dv}}}}{{\rm{v}}}}} = \frac{{ - {\rm{dP}}}}{{{\rm{dv}}}} \times {\rm{V}}\)      ----(i)

For isothermal process:

\(\frac{{\rm{P}}}{{\rm{\rho }}} = {\rm{Constant}} \Rightarrow {\rm{P}} \times {\rm{V}} = {\rm{constant}}\)      ----(ii)

Differentiating equation (ii),

PdV + Vdp = 0

⇒ PdV = -Vdp

\(\Rightarrow {\rm{P}} = \frac{{ - {\rm{VdP}}}}{{{\rm{dV}}}}\)      ----(iii)

From equation (i) & (iii), we have

K = P

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For adiabatic condition, \(\frac{{\rm{P}}}{{{{\rm{\rho }}^{\rm{k}}}}} = \) constant, where k = Ratio of specific heats.

Bulk modulus, K = Pk

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