Question
Download Solution PDFIn the isothermal condition, the isothermal bulk modulus of an ideal gas is equal to ______.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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
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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