When the pressure p on a material increases by an amount ?p, the volume of the material will change from V to V + ?V, where ?V is negative. The bulk modulus B of the material is defined to be the ratio of the pressure change ?p to the absolute value |?V/V| of the fractional volume change. The greater the bulk modulus, the greater the pressure increase required for a given fractional volume change, and the more incompressible the material (see Section 11.4). Since ?V (a) Use the result of Problem 42.53 to show that the bulk modulus for a system of N free electrons in a volume V at low temperatures is B = 5/3p.(b) Evaluate the bulk modulus for the electrons in copper, which has a free-e1ectron concentration of 8.45 X 1028 m-3. Express your result in Pascal??s.(c) The actual bulk modulus of copper is 1.4 x 1011 Pa. Based on your result in part (b), what fraction of this is due to the free electrons in copper? (This result shows that the free electrons in a metal play a major role in making the metal resistant to compression.) What do you think is responsible for the remaining fraction of the bulk modulus?

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dp dV B = -V