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Mechanical Properties Of Fluids

Question
CBSEENPH11018091

Show that isothermal bulk modulus of a gas is equal to the pressure of the gas.

Solution
Bulk modulus is given by,
                   straight K space equals space minus straight V space dP over dV
Negative sign implies if pressure increases volume decreases.
For isothermal condition, equations of state is given by, 
                     PV = constant 
Now, differentiating this equation, we get 
space space space space space space space Pdv space plus space VdP space equals space 0

rightwards double arrow space space dP over dV space equals space minus straight P over straight V
space space space space space space space space space space straight K space equals space minus space straight V space dP over dV
space space space space space space space space space space space space space space equals space minus space straight V space dP over dV space
space space space space space space space space space space space space space space equals space minus straight V space left parenthesis negative straight P over straight V right parenthesis
space space space space space space space space space space space space space space equals space straight P space
space space space space space space space space space space space space
Therefore, bulk modulus of a gas is equal to the pressure of a gas is proved.