Problem 129 - Chemistry

Question

How many photons at \(660 \mathrm{nm}\) must be absorbed to melt \(5.0 \times 10^{2} \mathrm{~g}\) of ice \(?\) On average, how many \(\mathrm{H}_{2} \mathrm{O}\) molecules does one photon convert from ice to water? (Hint: It takes \(334 \mathrm{~J}\) to melt \(1 \mathrm{~g}\) of ice at \(0^{\circ} \mathrm{C}\).)

Step-by-Step Solution

Answer

To melt \(5.0 \times 10^{2} \mathrm{~g}\) of ice, it requires a total number of photons that sorted from detailed steps. On average, one photon converts a specific number of H2O molecules from ice to water, which can be calculated by finding the ratio between total water molecules in the ice and the number of photons required.

Step 1: Calculate energy per photon

First needed component is the energy of one photon. According to Planck's formula, the energy of a photon is given by \(E=hc/λ\), where h is Planck's constant (\(6.63 \times 10^{-34} \mathrm{~J \cdot s})\), c is the speed of light (\(3.0 \times 10^{8} \mathrm{~m/s})\), and λ is the wavelength of light. Substitute the provided wavelength value, \(660 \times 10^{-9}\) \(\mathrm{~m}\), into the formula to find the energy of a photon.

Step 2: Calculate total energy needed

Next, calculate the total energy needed to melt the ice using the formula \(Q=mcδT\), where m is the mass, c is the specific heat, δT is the change in temperature, and Q is the heat transferred. Given that it takes \(334 \mathrm{~J}\) to melt \(1 \mathrm{~g}\) of ice, the total energy needed to melt \(5.0 \times 10^{2}\) \( \mathrm{~g}\) of ice would be \(334 \mathrm{~J/g} \times 5.0 \times 10^{2}\) \(\mathrm{~g}\).\n\n

Step 3: Calculate required number of photons

Next, divide the total energy required to melt the ice by the energy of a single photon to find the number of photons required.\n\n

Step 4: Determine molecules per photon

Finally, calculate the number of H2O molecules affected by one photon. One mole (or \(6 \times 10^{23} \)) of water molecules weighs about \(18 \mathrm{~g}\). Thus, in \(5.0 \times 10^{2} \mathrm{~g}\) of ice there are roughly \(1.7 \times 10^{25} \) water molecules. Dividing this by the number of photons gives the average number of molecules affected by one photon.

Key Concepts

Photon Energy Calculation

Understanding photon energy is key when we discuss phenomena like melting ice with light. The energy of a photon is calculated using the equation \(E = \frac{hc}{\lambda}\), where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \mathrm{J \cdot s}\)), \(c\) is the speed of light (\(3.0 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the wavelength of the photon. For a photon with a wavelength of 660 nm, which is in the visible part of the spectrum, this formula lets us determine how much energy each photon carries.
To melt ice, the absorbed photons must have enough energy to break the molecular bonds holding the ice's structure together. Planck's equation gives us a way to quantify that energy and use it to calculate how many photons are needed to melt a specific quantity of ice.

Specific Heat and Latent Heat

To grasp how ice melts, we must comprehend the concepts of specific heat and latent heat. Specific heat is the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius. Conversely, latent heat refers to the heat required to change the state of a substance without changing its temperature.
In the context of melting ice, the relevant form of latent heat is the latent heat of fusion, which is the heat required to transform ice into water at 0°C without changing the temperature. This value for water is 334 J/g. It's essential because the energy contributed by the photon isn't just incrementing temperature; it's effecting a change of state, a phase transition from solid to liquid. By knowing the mass of the ice and the latent heat, we can calculate the total amount of energy needed to melt it.

Avogadro's Number and Mole Concept

To link the microscopic world of photons and molecules to the macroscopic quantities we work with every day, we use Avogadro's number and the mole concept. A mole is a unit in chemistry that represents a specific number of particles, namely \(6.022 \times 10^{23}\) particles, known as Avogadro's number. This allows us to relate mass to the number of molecules.
When we say that water has a molar mass of 18 g/mol, we mean that one mole (or Avogadro's number of molecules) of water weighs 18 grams. With this in mind, we can determine the number of water molecules in a given mass of ice. Then by dividing the total number of water molecules by the number of photons required to melt the ice, we can find out on average how many H\textsubscript{2}O molecules are affected by a single photon. This bridges the gap between the fundamental particle interactions and the observable melting of the ice.