What is the mass of photon of sodium light having a wavelength of 5890A˚ ? (Given h=6.6×`10^(−27)`ergsec)

Options

A. 3.735×`10^(−33)`g
B. 3.735×`10^(−29)`g
C. 3.735×`10^(−31)`g
D. 1.57×`10^(−33)`g

Answer

To find the mass of a photon of sodium light with a wavelength of 5890 Å, we can use the relationship derived from the de Broglie hypothesis. Here’s a step-by-step solution:

### Step 1: Convert Wavelength from Angstroms to Centimeters
The wavelength is given as 5890 Å. We need to convert this to centimeters.

1 Å = \(10^{-8}\) cm

So,
\[ 5890 \, \text{Å} = 5890 \times 10^{-8} \, \text{cm} = 5.89 \times 10^{-5} \, \text{cm} \]

### Step 2: Identify the Speed of Light
The speed of light (v) is a constant value. In centimeters per second, it is:
\[ v = 3 \times 10^{10} \, \text{cm/s} \]

### Step 3: Use the Formula for Mass of Photon
The mass (m) of a photon can be calculated using the formula:
\[ m = \frac{h}{\lambda \cdot v} \]
where:
- \(h\) is Planck's constant, given as \(6.6 \times 10^{-27} \, \text{erg sec}\),
- \(\lambda\) is the wavelength in cm,
- \(v\) is the speed of light in cm/s.

### Step 4: Substitute the Values into the Formula
Now, substituting the values into the formula:
\[ m = \frac{6.6 \times 10^{-27} \, \text{erg sec}}{(5.89 \times 10^{-5} \, \text{cm}) \cdot (3 \times 10^{10} \, \text{cm/s})} \]

### Step 5: Calculate the Denominator
First, calculate the denominator:
\[ 5.89 \times 10^{-5} \, \text{cm} \cdot 3 \times 10^{10} \, \text{cm/s} = 1.767 \times 10^{6} \, \text{cm}^2/\text{s} \]

### Step 6: Calculate the Mass
Now, substitute this back into the mass equation:
\[ m = \frac{6.6 \times 10^{-27}}{1.767 \times 10^{6}} \approx 3.735 \times 10^{-33} \, \text{grams} \]

### Final Answer
The mass of the photon of sodium light having a wavelength of 5890 Å is approximately:
\[ \boxed{3.735 \times 10^{-33} \, \text{grams}} \]