Question

How many photons at \(586 \mathrm{nm}\) 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 \(0^{\circ} \mathrm{C}\).)

Short answer

Approximately \(4.93 \times 10^{23}\) photons are needed, and each converts about 34 water molecules.

Step-by-step solution

Calculate the Total Energy to Melt the Ice

First, we need to determine the total energy required to melt the ice. We use the formula \( q = m \times \, \Delta H_f \), where \( q \) is the energy in joules, \( m \) is the mass of the ice, and \( \, \Delta H_f \) is the heat of fusion. Given that \( \, \Delta H_f = 334 \, \text{J/g} \) and \( m = 5.0 \times 10^2 \, \text{g} \),\[q = 5.0 \times 10^2 \, \text{g} \times 334 \, \text{J/g} = 167,000 \, \text{J}\].So, the total energy needed is \( 167,000 \, \text{J} \).

Calculate the Energy of One Photon

The energy of a single photon is given by the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \(6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \), \( c \) is the speed of light \(3.00 \times 10^8 \, \text{m/s} \), and \( \lambda \) is the wavelength in meters. For \( \lambda = 586 \, \text{nm} = 586 \times 10^{-9} \, \text{m} \),\[E = \frac{6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \times 3.00 \times 10^8 \, \text{m/s}}{586 \times 10^{-9} \, \text{m}} \approx 3.39 \times 10^{-19} \, \text{J}.\]So, each photon's energy is approximately \( 3.39 \times 10^{-19} \, \text{J} \).

Calculate the Number of Photons Required

To find the number of photons needed, divide the total energy by the energy of one photon:\[\text{Number of photons} = \frac{167,000 \, \text{J}}{3.39 \times 10^{-19} \, \text{J/photon}} \approx 4.93 \times 10^{23}.\]Thus, approximately \( 4.93 \times 10^{23} \) photons are required to melt the ice.

Calculate the Number of Water Molecules in the Ice

We know from Avogadro's number that one mole of water contains \( 6.022 \times 10^{23} \) molecules. First, calculate the moles of water:\[\text{Moles of water} = \frac{500 \, \text{g}}{18.015 \, \text{g/mol}} \approx 27.76 \, \text{mol}.\]Now, calculate the number of water molecules:\[\text{Number of } \text{H}_2\text{O} \text{ molecules} = 27.76 \, \text{mol} \times 6.022 \times 10^{23} \, \text{molecules/mol} \approx 1.67 \times 10^{25}.\]

Find the Average Number of Water Molecules Melted by One Photon

To find how many water molecules each photon converts:\[\text{Average molecules per photon} = \frac{1.67 \times 10^{25} \, \text{molecules}}{4.93 \times 10^{23} \, \text{photons}} \approx 34.\]On average, one photon converts about 34 water molecules from ice to water.

Key concepts

Ice Melting Energy

When it comes to melting ice, energy is a key player. To understand this process, we calculate how much energy is required for conversion. It involves understanding the intrinsic energy stored in the form of heat, needed to change solid ice into liquid water at a constant temperature.

This energy, called the heat of fusion, is specific to water and represents the amount needed to change 1 gram of ice at 0°C into water without altering the temperature. In our problem, melting 500 grams of ice requires multiplying the heat of fusion (334 J/g) by the total mass of ice (500 g), totaling 167,000 Joules.

So, the ice melting energy is the total energy necessary for this phase transition to happen, which in this case, is 167,000 Joules.

Planck's Constant

Planck's constant is a fundamental figure in physics, playing a critical role in the field of quantum mechanics. It represents the constant relationship between energy and frequency for photons. This constant, denoted by 'h', has a value of approximately \(6.626 \times 10^{-34} \, \text{J} \cdot \text{s}\).

• Use it to calculate the energy of photons based on their wavelength or frequency.
• Serves as a bridge connecting macroscopic and quantum phenomena.
• Essential in formulas calculating photon-related energies.

For the problem of calculating the energy of a photon with a wavelength of 586 nm, Planck's constant is a vital component. The energy calculation uses the formula \( E = \frac{hc}{\lambda} \), linking Planck's constant with the wavelength and speed of light. This helped us determine that each photon at this wavelength carries approximately \(3.39 \times 10^{-19} \, \text{J}\) of energy.

Avogadro's Number

Avogadro's number is a fundamental constant that helps bridge atomic scale measurements to macroscopic quantities. Denoted as \(6.022 \times 10^{23}\) mol⁻¹, it represents the number of individual units (atoms, molecules, etc.) in one mole of a substance, like water.

• Facilitates the conversion from moles to molecules or atoms.
• Key for calculations in both chemistry and physics.
• Allows us to quantify microscopic scale reactions in lab-scale quantities.

In our exercise, knowing the moles of water involved, Avogadro's number was essential to determine the total number of water molecules available for potential conversion from ice to liquid. Knowing there are 27.76 moles of water, Avogadro's number allowed us to calculate approximately \(1.67 \times 10^{25}\) molecules of \(\text{H}_2\text{O}\) in the ice.

Heat of Fusion

The heat of fusion is the amount of energy required to change a substance from solid to liquid at its melting point, without changing the temperature. For water, this value is 334 Joules per gram. This specific heat is a crucial quantity in thermodynamics and explains how substances transition between phases.

• Unique for each substance and varies based on its molecular structure.
• Represents energy needed to overcome the bonds holding the molecules in a solid state.
• Important for calculating energy requirements for melting processes.

In particular, the heat of fusion played a major role in our headlining problem by determining the amount of energy required to melt the given mass of ice. By knowing this property, we could compute the energy needed for the phase change from ice to water, ensuring precise calculation for the photons absorbed during this transformation.