Assuming that radiation with λ=15.0 cm is used, and that all the energy is converted to heat, with 4.184 J needed to raise the temperature of 1.00 g of water by 1.00 °C, how many photons are necessary to raise the temperature of a 300 g sample of water by 1.00 °C?

• A. 1.20 x 10^23 photons
• B. 5.00 x 10^21 photons
• C. 3.00 x 10^24 photons
• D. 2.50 x 10^22 photons

Answer: (A)

Verified step by step guidance

1. First, calculate the total energy required to raise the temperature of the 300 g sample of water by 1.00 °C. Use the formula: \( E = m \cdot c \cdot \Delta T \), where \( m \) is the mass of the water (300 g), \( c \) is the specific heat capacity of water (4.184 J/g°C), and \( \Delta T \) is the change in temperature (1.00 °C).
2. Next, determine the energy of a single photon using the wavelength \( \lambda = 15.0 \) cm. Use the equation \( E_{photon} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \) J·s, \( c \) is the speed of light \( 3.00 \times 10^{8} \) m/s, and \( \lambda \) is the wavelength converted to meters (0.15 m).
3. Calculate the energy of a single photon using the values from the previous step. Substitute \( h \), \( c \), and \( \lambda \) into the equation \( E_{photon} = \frac{hc}{\lambda} \) to find the energy of one photon.
4. Determine the number of photons required by dividing the total energy needed to raise the temperature of the water by the energy of a single photon. Use the formula: \( N = \frac{E_{total}}{E_{photon}} \), where \( E_{total} \) is the energy calculated in the first step and \( E_{photon} \) is the energy of a single photon calculated in the third step.
5. Finally, compare the calculated number of photons \( N \) with the given options to identify the correct answer.