Question

Assume that water absorbs light of wavelength \(4.20 \times 10^{-6} \mathrm{m}\) with \(100 \%\) efficiency. How many photons are required to heat \(5.75 \mathrm{g}\) of water by \(1.00 \mathrm{K} ?\) The heat capacity of water is \(75.3 \mathrm{Jmol}^{-1} \mathrm{K}^{-1}\)

Short answer

The number of photons required to heat 5.75 g of water by 1.00 K can be calculated as follows: 1. Calculate the number of moles of water: \(n = \frac{5.75\,\text{g}}{18.01\,\text{g/mol}} = 0.319\,\text{mol}\) 2. Calculate the heat required to heat the water by 1.00 K: \(q = n \times C_p \times \Delta T = 0.319\, \text{mol} \times 75.3\, \text{J}\text{mol}^{-1}\text{K}^{-1} \times 1.00\, \text{K} = 24.01\, \text{J}\). 3. Calculate the energy per photon: \(E = \frac{hc}{\lambda} = \frac{6.63 \times 10^{-34}\, \text{Js} \times 3.00 \times 10^{8}\, \text{m/s}}{4.20 \times 10^{-6}\, \text{m}} = 4.74 \times 10^{-19}\, \text{J}\). 4. Calculate the number of photons needed: \(N_{\text{photons}} = \frac{q}{E} = \frac{24.01\, \text{J}}{4.74 \times 10^{-19}\, \text{J}} = 5.07 \times 10^{20}\) photons. Thus, \(5.07 \times 10^{20}\) photons are required to heat 5.75 g of water by 1.00 K.

Step-by-step solution

Calculate the heat required to heat the water by 1ºK

First, we need to calculate the number of moles of water in 5.75 g of water. The molecular weight of water (H2O) is 18.01 g/mol. To calculate the number of moles, we will use the formula: Moles of water = mass (g) / molecular weight (g/mol) Once we have the number of moles of water, we can use the heat capacity of water and the given temperature change to find the heat required using the following relationship: Heat required (q) = number of moles × heat capacity × ΔT where ΔT is the change in temperature (1.00 K in this case).

Calculate the energy per photon

Next, we will calculate the energy per photon using the provided wavelength and the formula: Energy per photon (E) = \(h\times c / \lambda\) where h is Planck's constant (6.63 x 10^{-34} Js), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength of absorbed light (4.20 x 10^{-6} m).

Calculate the number of photons needed

Now that we have the heat required to heat the water and the energy of one absorbed photon, we can calculate the number of photons required to heat the water by the given temperature using the formula: Number of photons = Heat required / Energy per photon Make sure to use the appropriate units for all the calculations.

Key concepts

Heat Capacity

Heat capacity is a property that measures the amount of heat energy required to raise the temperature of a substance by a certain amount, namely one degree Celsius or one Kelvin. In simpler terms, it quantifies how much heat a material can 'hold' or 'soak up'. For water, it's particularly high, meaning that water can absorb a lot of heat before its temperature changes significantly. When we talk about the heat capacity of water in a calculation, we are often referring to its specific heat capacity or molar heat capacity.

In the context of the exercise, the molar heat capacity of water is given as \(75.3 \text{Jmol}^{-1} \text{K}^{-1}\). This means that to raise the temperature of 1 mole of water by 1 K (or 1 °C), 75.3 joules of energy is needed. This value is essential for determining how much energy, and therefore how many photons, are needed to heat a given amount of water by a certain temperature change.

Moles of Substance

Moles of a substance refer to the amount of that substance in terms of the number of particles, like atoms or molecules, it has. One mole is defined as the amount of substance that contains as many particles as there are atoms in 12 grams of pure carbon-12. This number is called Avogadro's number, which is approximately \(6.022 \times 10^{23}\).

When working with substances and their reactions or interactions, chemists often measure in moles because it's a convenient way to link the mass of a substance to the number of particles it contains. In our exercise, the mass of water is given, and the solution requires calculating the moles of water to find out the total amount of heat needed to raise its temperature. With the molecular weight of water being 18.01 g/mol, the conversion from grams to moles becomes a simple division.

Energy per Photon

The energy per photon concept relates to how much energy is carried by a single photon, which is the smallest discrete amount or quantum of electromagnetic radiation. This energy is inversely proportional to the photon's wavelength; photons with shorter wavelengths have higher energies, and vice versa.

In the solution, to calculate the energy of a photon, the formula \(E = h \times \frac{c}{\lambda}\) is used, where \(E\) is the energy of the photon, \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(h\) is Planck's constant. For our problem, knowing the energy per photon is crucial because it will determine how many photons are needed to transfer the necessary amount of heat to the water.

Planck's Constant

Finally, Planck's constant (denoted as \(h\)) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. The value of this constant is approximately \(6.626 \times 10^{-34}\) joule-seconds. Planck's constant is a critical factor in the quantization of energy levels within atoms and molecules and thereby plays a pivotal role in the field of quantum physics.

In the context of our problem, Planck's constant is part of the formula used to determine the energy of each photon. As every photon's energy level is quantized, this constant helps bridge the world of quantum mechanics with classical physical phenomena, such as heating water, because it provides a way to equate a countable number of photons with a measurable amount of heat energy.