For various stable nuclei, a graph is plotted with the natural logarithm of the nuclear surface area ( S ) on the y -axis and the natural logarithm of the mass number ( A ) on the x -axis. The resulting graph is a straight line whose slope is p q , where p and q are coprime positive integers. The value of 10p + q is _______.

Step-by-step solution

The radius of a nucleus depends on its mass number A as R = R₀ A^ 1/3 . The surface area of the nucleus is given by S = 4 R^2 . Substituting the expression for R gives: S = 4 (R₀ A^ 1/3 )^2 = 4 R₀^2 A^ 2/3 Taking the natural logarithm on both sides yields: S = (4 R₀^2) + 2 3 A Comparing this with the equation of a straight line y = mx + c , where y = S and x = A , the slope m is 2 3 . Given that the slope is p q in lowest terms, we have p = 2 and q = 3 . Therefore, the value of 10p + q is 10(2) + 3 = 23 . Answer: 2