Answer :
To calculate the wavelength of the X-rays using Bragg's Law, follow these steps:
1. Understand Bragg's Law: Bragg's Law can be described by the equation [tex]\( n\lambda = 2d \sin(\theta) \)[/tex], where:
- [tex]\( n \)[/tex] is the order of reflection (given as 1).
- [tex]\( \lambda \)[/tex] is the wavelength of the X-rays.
- [tex]\( d \)[/tex] is the distance between the crystal planes.
- [tex]\( \theta \)[/tex] is the glancing angle.
2. Given data:
- [tex]\( d = 99.3 \)[/tex] pm (picometers)
- [tex]\( \theta = 10.42^\circ \)[/tex]
- [tex]\( n = 1 \)[/tex]
3. Convert the angle to radians:
- Since trigonometric functions in mathematical equations typically require angles in radians, convert [tex]\( \theta \)[/tex] from degrees to radians. The formula for conversion is:
[tex]\[
\text{Radians} = \theta \times \left(\frac{\pi}{180}\right)
\][/tex]
- After conversion, the angle [tex]\( \theta \)[/tex] in radians is approximately 0.1819 radians.
4. Calculate the wavelength:
- Substitute the known values into Bragg's Law:
[tex]\[
\lambda = \frac{2d \sin(\theta)}{n}
\][/tex]
- Plug in the values:
- [tex]\( n = 1 \)[/tex]
- [tex]\( d = 99.3 \)[/tex] pm
- [tex]\( \sin(\theta) = \sin(0.1819) \)[/tex]
- Calculate the sine of the angle and then apply it to the formula to find [tex]\( \lambda \)[/tex].
- The calculated wavelength [tex]\( \lambda \)[/tex] is approximately 35.92 picometers.
By following these steps, we determined that the wavelength of the X-rays used is approximately 35.92 pm.
1. Understand Bragg's Law: Bragg's Law can be described by the equation [tex]\( n\lambda = 2d \sin(\theta) \)[/tex], where:
- [tex]\( n \)[/tex] is the order of reflection (given as 1).
- [tex]\( \lambda \)[/tex] is the wavelength of the X-rays.
- [tex]\( d \)[/tex] is the distance between the crystal planes.
- [tex]\( \theta \)[/tex] is the glancing angle.
2. Given data:
- [tex]\( d = 99.3 \)[/tex] pm (picometers)
- [tex]\( \theta = 10.42^\circ \)[/tex]
- [tex]\( n = 1 \)[/tex]
3. Convert the angle to radians:
- Since trigonometric functions in mathematical equations typically require angles in radians, convert [tex]\( \theta \)[/tex] from degrees to radians. The formula for conversion is:
[tex]\[
\text{Radians} = \theta \times \left(\frac{\pi}{180}\right)
\][/tex]
- After conversion, the angle [tex]\( \theta \)[/tex] in radians is approximately 0.1819 radians.
4. Calculate the wavelength:
- Substitute the known values into Bragg's Law:
[tex]\[
\lambda = \frac{2d \sin(\theta)}{n}
\][/tex]
- Plug in the values:
- [tex]\( n = 1 \)[/tex]
- [tex]\( d = 99.3 \)[/tex] pm
- [tex]\( \sin(\theta) = \sin(0.1819) \)[/tex]
- Calculate the sine of the angle and then apply it to the formula to find [tex]\( \lambda \)[/tex].
- The calculated wavelength [tex]\( \lambda \)[/tex] is approximately 35.92 picometers.
By following these steps, we determined that the wavelength of the X-rays used is approximately 35.92 pm.
