The binding energy of electrons in a metal is 194 kJ/mol.

Find the threshold frequency of the metal.

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The heat of fusion of ice is 6.00 kJ/mol.

Find the number of photons of wavelength = 6.78×10^−6 m that must be absorbed to melt 6.80 g of ice.

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To find the threshold frequency of the metal, we can use the equation:

E = hf

Where E is the energy, h is Planck's constant (6.626 x 10^-34 J.s), and f is the frequency.

We can relate the energy to the binding energy using the equation:

E = BE/n

Where BE is the binding energy and n is the Avogadro's number (6.022 x 10^23 mol^-1).

Plugging in the values and solving for f, we have:

BE = 194 kJ/mol = (194 x 10^3 J/mol) = (194 x 10^3 J/mol) / (6.022 x 10^23 mol^-1)

Now, we have the value of E (energy), and we can rearrange the equation E = hf to solve for f:

f = E/h

Plugging in the values, we have:

f = (194 x 10^3 J/mol) / (6.022 x 10^23 mol^-1) / (6.626 x 10^-34 J.s)

Simplifying the expression, we find the threshold frequency of the metal.

For the second question, to find the number of photons required to melt a certain amount of ice, we can use the equation:

E = nhf

Where E is the energy, n is the number of photons, h is Planck's constant, and f is the frequency.

First, we need to find the energy required to melt the ice using the heat of fusion:

E = heat of fusion x number of moles of ice

Given the heat of fusion (6.00 kJ/mol) and the mass of ice (6.80 g), we can convert the mass of ice to moles using the molar mass of water (18.015 g/mol):

Number of moles of ice = (6.80 g) / (18.015 g/mol)

Now, we have the value of E (energy), and we can rearrange the equation E = nhf to solve for n:

n = E / (hf)

To find the frequency, we can use the equation:

c = λf

Where c is the speed of light (3.00 x 10^8 m/s), λ is the wavelength, and f is the frequency.

Given the wavelength (6.78 x 10^-6 m), we can rearrange the equation to solve for f:

f = c / λ

Plugging in the values, we have:

f = (3.00 x 10^8 m/s) / (6.78 x 10^-6 m)

Now we have the values of E (energy) and f (frequency), we can calculate the number of photons required to melt the ice by plugging them into the equation n = E / (hf).