Final answer:
Approximately 5.431 × 10^25 photons with a wavelength of 2.5 mm must be absorbed to heat 1 gram of water to 100 degrees Celsius.
Step-by-step explanation:
To calculate the number of photons that must be absorbed to heat 1 gram of water to 100 degrees Celsius, we need to use the equation E = mcΔT, where E is the energy absorbed, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature. The specific heat capacity of water is approximately 4.18 J/g·°C.
First, we calculate the energy required to heat the water:
E = (1 g)(4.18 J/g·°C)(100 °C) = 418 J
Next, we need to convert the energy to the number of photons by using the equation E = nhν, where n is the number of photons, h is Planck's constant (6.626 × 10^-34 J·s), and ν is the frequency of the photons.
Let's calculate the frequency:
ν = c/λ, where c is the speed of light (3.00 × 10^8 m/s) and λ is the wavelength of the photons (2.5 mm or 2.5 × 10^-3 m).
ν = (3.00 × 10^8 m/s)/(2.5 × 10^-3 m) = 1.2 × 10^11 Hz
Now, we can calculate the number of photons:
418 J = n(6.626 × 10^-34 J·s)(1.2 × 10^11 Hz)
n = 418/(6.626 × 10^-34 J·s)(1.2 × 10^11 Hz) ≈ 5.431 × 10^25 photons
Therefore, approximately 5.431 × 10^25 photons with a wavelength of 2.5 mm must be absorbed to heat 1 gram of water to 100 degrees Celsius.