The compressibility k of a substance is defined as the fractional change in volume of that substance for a given changein pressure:                              k = - 1/V dV/dP(a) Explain why the negative sign in this expression ensures k is always positive. (b) Show that if an ideal gas is compressed isothermally, its compressibility is given by k1 = 1/P. (c) What If? Show that if an ideal gas is compressed adiabatically, its compressibility is given by k2 = 1/(γi). Determine values for (d) k1 and (e) k2 for a monatomic ideal gas at a pressure of 2.00 atm.

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