Following results are observed when sodium metal is irradiated with different wavelengths. Calculate (a) threshold wavelength and (b) Planck's constant.
`{:(lambda(nm),,5000,450,400),(vxx10^(-5)(cm s^(-1)),,2.55,4.35,5.35):}`

Explanation

(a) Assuming the threshold wavelength to be `lambda_(0) nm (= lambda_(0)xx10^(9)m)` , the kinetic energy of the radiation is given as:
`h(v-V_(0))=(1)/(2)mv^(2)`
Three different equalities can be formed by the given value as :
` hc((1)/(lambda)-(1)/(lambda_(0)))=(1)/(2) mv^(2)`
`hc((1)/(500xx10^(9))-(1)/(lambda_(0)x10^(-9)m))=(1)/(2) (2.55xx10^(+5)xx10^(-2)ms^(-1))`
`(hc)/(10^(-9)m)[(1)/(500)-(1)/(lambda_(0))]=(1)/(2)m (2.55xx10^(+5) ms^(-1))^(2)" " (1)`
similarly,
`(hc)/(10^(-9)m)[(1)/(450)-(1)/(lambda_(0))]=(1)/(2)m (3.45xx10^(+5) ms^(-1))^(2)" " (2)`
`(hc)/(10^(-9)m)[(1)/(400)-(1)/(lambda_(0))]=(1)/(2)m (5.35xx10^(+5) ms^(-1))^(2)" " (3)`
Dividing equation (3) by equation (1):
`([(lambda_(0)-400)/(400 lambda_(0))]]/([(lambda_(0)-500)/(500 lambda_(0))])=((5.35xx10^(+3)ms^(-1))^(2))/((2.55xx10^(+3)ms^(-1))^(2))`
`(5 lambda_(0)-2000)/(4 lambda_(0)-2000)=((5.35)/(2.55))^(2)=(28.6225)/(6.5025)`
`(5 lambda_(0)-2000)/(4 lambda_(0)-2000)=4.40177`
`17.6070 lambda_(0)=5 lambda_(0)=8803.537-2000`
`lambda_(0)=(6805.537)/(12.607)`
`lambda _(0)=539.8 nm`
`lambda_(0)=540 nm`
So, threshold wavelength`(lambda_(0)) = 540 nm`
Note: part (b) of the question is not done due to the incorrect values of velocity given in the question.
`5 lambda_(0)-2000/(4 lambda_(0)-2000)=((5.35)/(2.55))^(2)=(28.6225)/(6.5025)`
`5 lambda_(0)-2000)/(4 lambda_(0)-2000)=4.40177`
`17.6070 lambda_(0)=5 lambda_(0)=8803.537-2000`
`lambda_(0)=(6805.537)/(12.607)`
`lambda _(0)=539.8 nm`
`lambda_(0)=540 nm`