Assuming that radiation with λ = 15.0 cm is used, that all the energy is converted to heat, and that 4.184 J is 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 350 g sample of water by 1.00 °C?

• A. 1.00 x 10^23 photons
• B. 4.00 x 10^23 photons
• C. 3.00 x 10^23 photons
• D. 2.00 x 10^23 photons

Answer: (C)

Verified step by step guidance

1. First, calculate the energy required to raise the temperature of the 350 g water sample by 1.00 °C. Use the formula: \( q = m \cdot c \cdot \Delta T \), where \( q \) is the heat energy, \( m \) is the mass of the water (350 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, calculate the energy of a single photon using the formula: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (6.626 \times 10^{-34} \text{ J s}), \( c \) is the speed of light (3.00 \times 10^8 \text{ m/s}), and \( \lambda \) is the wavelength of the radiation (15.0 cm, converted to meters as 0.15 m).
3. Determine the total energy required to raise the temperature of the water by multiplying the mass of the water by the specific heat capacity and the temperature change: \( q = 350 \text{ g} \times 4.184 \text{ J/g°C} \times 1.00 \text{ °C} \).
4. Calculate the number of photons needed by dividing the total energy required by the energy of a single photon: \( \text{Number of photons} = \frac{q}{E} \).
5. Finally, compare the calculated number of photons to the given options to determine the correct answer.