Question

The isothermal bulk modulus of a perfect gas at atmospheric pressure is-
1. 1.013 × 10⁵ N/m²
2. 1.013 × 10⁶ N/m²
3. 1.013 × 10¹⁰ N/m²
4. 1.013 × 10¹¹ N/m²

Answer 1

Correct Answer - Option 1 : 1.013 × 10⁵ N/m²

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

The magnitude of atmospheric pressure is 1.013 × 105 N/m2

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

Bulk modulus, K = γP