Let's break down the problem step by step: --- ### **Step 1: Identify the Formula** The problem gives us the equation for the energy of a photon: \[ E = h \nu \] Here, \( h \) is Planck's constant (\( 6.626 \times 10^{-34}\ \text{J·s} \)), and \( \nu \) (nu) is the frequency. **Explanation:** This formula tells us that the energy of a photon is directly proportional to its frequency. --- ### **Step 2: Substitute the Given Values** From the image, you are given: - \( h = 6.626 \times 10^{-34}\ \text{J·s} \) - \( \nu = 5.10 \times 10^{14}\ \text{s}^{-1} \) Substitute these into the formula: \[ E = (6.626 \times 10^{-34}\ \text{J·s}) \times (5.10 \times 10^{14}\ \text{s}^{-1}) \] **Explanation:** We are directly substituting the values for Planck's constant and frequency into the equation to find the energy. --- ### **Step 3: Calculate the Energy** Multiply the numbers: \[ E = (6.626 \times 5.10) \times 10^{-34 + 14}\ \text{J} \] \[ E = 33.7926 \times 10^{-20}\ \text{J} \] Now, adjust to scientific notation: \[ E = 3.38 \times 10^{-19}\ \text{J} \] (Rounded to 3 significant figures) **Explanation:** First, we multiply the decimal numbers, then add the exponents for powers of 10, and finally express the answer in correct scientific notation. --- ## **Final Answer:** \[ \boxed{E = 3.38 \times 10^{-19}\ \text{J}} \]

The concept used in solving this problem is based on Planck's equation, which states that the energy of a photon is directly proportional to its frequency, mathematically expressed as \( E = h \nu \). Here, \(h\) is Planck's constant, a fundamental constant in quantum physics, and \(\nu\) is the frequency of the electromagnetic wave. By substituting the given values of \(h\) and \(\nu\) into the equation, we calculate the photon’s energy. This approach highlights the quantum nature of light, where energy is quantized and depends on the wave’s frequency.