Question

Derive expression for the radius of the nth bohr orbit in an atom.hence show that the radius of the orbit is directly proportional to the square of the principal quantum number​

Answer 1

we have to drive the expression for the radius of the nth Bohr's orbit in an atom. and also show that the radius of the orbit is directly proportional to the square of the principal quantum number.

solution : according to Bohr's atomic theory,

electrostatic force between nucleus and an electron in nth orbit is equal to centripetal force acting on electron.

i.e.,  $\frac{ze^2}{4\pi\epsilon_0r^2}=\frac{mv^2}{r}$  

⇒      $r=\frac{Ze^2}{4\pi\epsilon_0mv^2}$  ...(1)

now angular momentum of an electron in nth orbit is integral multiple of h/2π.

$mvr=\frac{nh}{2\pi}\\\implies v=\frac{nh}{2\pi mr}$   ...(2)

from equations (1) and (2) we get,

  $r=\frac{n^2h^2\epsilon_0}{\pi mZe^2}$ , this the expression ofthe radius of nth orbit in an atom.

from above expression, it is clear that r ∝ n² .

hence, the radius of the orbit is directly proportional to the square of the principal quantum number.  

Answer 2

Answer:

The graph of resistivity versus length should be a straight line with a negative slope, where the slope of the line is equal to the product of the resistance and cross-sectional area of the wire. The units of the slope will be ohm-meters squared, and the units of the x-axis will be meters (or any other appropriate unit of length).

Explanation:

The graph of resistivity (ρ) versus length (l) of a wire is typically a straight line if the wire has a uniform cross-sectional area and the resistivity is independent of temperature. This is known as the linear relationship between resistivity and length, given by the formula:

ρ = RA/l

where R is the resistance of the wire, A is its cross-sectional area, and l is its length.

Taking the reciprocal of both sides of the equation, we get:

1/ρ = l/RA

This equation has the form of a linear equation (y = mx + b), where 1/ρ is the dependent variable (y), l is the independent variable (x), and RA is the slope (m) of the line. The y-intercept (b) is zero, since the wire has zero length when the resistivity is infinite.

Therefore, the graph of resistivity versus length should be a straight line with a negative slope, where the slope of the line is equal to the product of the resistance and cross-sectional area of the wire. The units of the slope will be ohm-meters squared, and the units of the x-axis will be meters (or any other appropriate unit of length).

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