A graph is plotted showing the kinetic energy ( K ) of an electron on the y-axis versus the reciprocal of its orbital radius ( 1 r ) on the x-axis. Which of the following statements correctly describes the shape of the graphs and their relative slopes for a Hydrogen atom (H) and a singly ionized Helium ion (He ^+ )?

Step-by-step solution

The electrostatic force between the nucleus (charge Ze ) and the electron (charge e ) provides the centripetal force: mv^2 r = 1 4 ₀ Ze^2 r^2 Multiplying both sides by r , we get: mv^2 = 1 4 ₀ Ze^2 r The kinetic energy K of the electron is: K = 1 2 mv^2 = 1 8 ₀ Ze^2 r This can be written as: K = ( Ze^2 8 ₀ ) 1 r This is a linear equation of the form y = mx , where y = K and x = 1 r . Thus, the graph of K versus 1 r is a straight line passing through the origin. The slope of the graph is m = Ze^2 8 ₀ , which is dire