What is the number of photons of light with a wavelength of 4000 pm that provide 1J of energy?

Solution

Energy of one photon of light (E) = hv
Energy of n photons of light (E_n) = nhv
or $straight E subscript straight n space equals space straight n space straight h space straight c over straight lambda$
or $straight n space equals space fraction numerator straight E subscript straight n cross times straight lambda over denominator straight h cross times straight c end fraction$ ...(1)
Here, $straight E subscript straight n space equals space 1 space straight J$
$straight lambda space equals space 4000 space pm space equals space 4000 space cross times space 10 to the power of negative 12 end exponent straight m straight h space equals space 6.626 space cross times space 10 to the power of negative 34 end exponent Js semicolon space space straight c equals space space 3 cross times 10 to the power of 8 space ms to the power of negative 1 end exponent$
Substituting the value in eq. (1), we have,

$straight n space equals fraction numerator left parenthesis 1 space straight J right parenthesis space cross times space left parenthesis 4000 space cross times space 10 to the power of negative 12 end exponent straight m right parenthesis over denominator left parenthesis 6.626 space cross times 10 to the power of negative 34 end exponent Js right parenthesis cross times left parenthesis 3 cross times 10 to the power of 8 space ms to the power of negative 1 end exponent right parenthesis end fraction space space space space equals space fraction numerator 4000 space cross times space 10 to the power of negative 12 end exponent over denominator 6.626 space cross times 10 to the power of negative 34 end exponent cross times 3 cross times 10 to the power of 8 end fraction space space space space space equals 2 space cross times space 10 to the power of 16 space photons$