Answer :
To calculate the moment of inertia (I) of the [tex]$SiH_4$[/tex] ion when one of the hydrogen atoms is replaced by deuterium, we need to follow these steps:
### Step 1: Convert Bond Length
The bond length between Si and H is given as 147.98 pm (picometers).
[tex]\[ \text{Bond length in meters } (m) = 147.98 \times 10^{-12} \text{ meters} = 1.4798 \times 10^{-10} \text{ meters} \][/tex]
### Step 2: Determine the Atomic Masses
- Silicon (Si): Atomic mass of Silicon is 28.085 u.
[tex]\[ \text{Si mass in kg} = 28.085 \times 1.66053906660 \times 10^{-27} \text{ kg} = 4.6636239685461 \times 10^{-26} \text{ kg} \][/tex]
- Protium ( [tex]${}^1H$[/tex] ): Atomic mass of Protium is 1.00784 u.
[tex]\[ \text{H1 mass in kg} = 1.00784 \times 1.66053906660 \times 10^{-27} \text{ kg} = 1.673557692882144 \times 10^{-27} \text{ kg} \][/tex]
- Deuterium ( [tex]${}^2H$[/tex] ): Atomic mass of Deuterium is 2.014 u.
[tex]\[ \text{H2 mass in kg} = 2.014 \times 1.66053906660 \times 10^{-27} \text{ kg} = 3.344325680132399 \times 10^{-27} \text{ kg} \][/tex]
### Step 3: Calculate the Reduced Masses
Reduced mass [tex]\(\mu\)[/tex] for the system is defined as:
[tex]\[ \mu = \frac{m_1 \times m_2}{m_1 + m_2} \][/tex]
When calculating I for different configurations:
#### For [tex]$SiH_4$[/tex] with Protium ([tex]$H_1$[/tex]):
[tex]\[ \mu_{SiH1} = \frac{4 \, (\text{Si mass} \times \text{H1 mass})}{(\text{Si mass} + \text{H1 mass})} \][/tex]
[tex]\[ I_{H1} = \mu_{SiH1} \times (\text{Bond length})^2 \][/tex]
Putting in the values:
[tex]\[ I_{H1} = 4 \times \left( \frac{ (4.6636239685461 \times 10^{-26}) \times (1.673557692882144 \times 10^{-27})}{4.6636239685461 \times 10^{-26} + 1.673557692882144 \times 10^{-27}} \right) \times (1.4798 \times 10^{-10})^2 = 1.4151257562138218 \times 10^{-46} \text{ kg m}^2 \][/tex]
#### For [tex]$SiH_4$[/tex] with Deuterium ([tex]$H_2$[/tex]):
[tex]\[ \mu_{SiH1} = \frac{3 \, (\text{Si mass} \times \text{H1 mass})}{(\text{Si mass} + \text{H1 mass})} + \frac{\text{Si mass} \times \text{H2 mass}}{\text{Si mass} + \text{H2 mass}} \][/tex]
[tex]\[ I_{H2} = \left( 3 \times \left( \frac{ (4.6636239685461 \times 10^{-26}) \times (1.673557692882144 \times 10^{-27})}{4.6636239685461 \times 10^{-26} + 1.673557692882144 \times 10^{-27}} \right) + \frac{ (4.6636239685461 \times 10^{-26}) \times (3.344325680132399 \times 10^{-27})}{4.6636239685461 \times 10^{-26} + 3.344325680132399 \times 10^{-27}} \right) \times (1.4798 \times 10^{-10})^2 = 1.7446845178774643 \times 10^{-46} \text{ kg m}^2 \][/tex]
### Step 4: Calculate the Change in Moment of Inertia
[tex]\[ \Delta I = I_{H2} - I_{H1} \][/tex]
[tex]\[ \Delta I = 1.7446845178774643 \times 10^{-46} \text{ kg m}^2 - 1.4151257562138218 \times 10^{-46} \text{ kg m}^2 = 3.2955876166364254 \times 10^{-47} \text{ kg m}^2 \][/tex]
### Summary of Results
- The bond length is [tex]\(1.4798 \times 10^{-10} \)[/tex] meters.
- The atomic masses of Si, Protium, and Deuterium are [tex]\(4.6636239685461 \times 10^{-26}\)[/tex] kg, [tex]\(1.673557692882144 \times 10^{-27}\)[/tex] kg, and [tex]\(3.344325680132399 \times 10^{-27}\)[/tex] kg respectively.
- The moment of inertia [tex]\(I_{H1}\)[/tex] for [tex]\(\text{SiH}_4\)[/tex] with protium atoms is [tex]\(1.4151257562138218 \times 10^{-46}\)[/tex] kg m[tex]\(^2\)[/tex].
- The moment of inertia [tex]\(I_{H2}\)[/tex] for [tex]\(\text{SiH}_4\)[/tex] when one hydrogen atom is replaced with deuterium is [tex]\(1.7446845178774643 \times 10^{-46}\)[/tex] kg m[tex]\(^2\)[/tex].
- The change in moment of inertia when deuterium replaces protium is [tex]\(3.2955876166364254 \times 10^{-47}\)[/tex] kg m[tex]\(^2\)[/tex].
### Step 1: Convert Bond Length
The bond length between Si and H is given as 147.98 pm (picometers).
[tex]\[ \text{Bond length in meters } (m) = 147.98 \times 10^{-12} \text{ meters} = 1.4798 \times 10^{-10} \text{ meters} \][/tex]
### Step 2: Determine the Atomic Masses
- Silicon (Si): Atomic mass of Silicon is 28.085 u.
[tex]\[ \text{Si mass in kg} = 28.085 \times 1.66053906660 \times 10^{-27} \text{ kg} = 4.6636239685461 \times 10^{-26} \text{ kg} \][/tex]
- Protium ( [tex]${}^1H$[/tex] ): Atomic mass of Protium is 1.00784 u.
[tex]\[ \text{H1 mass in kg} = 1.00784 \times 1.66053906660 \times 10^{-27} \text{ kg} = 1.673557692882144 \times 10^{-27} \text{ kg} \][/tex]
- Deuterium ( [tex]${}^2H$[/tex] ): Atomic mass of Deuterium is 2.014 u.
[tex]\[ \text{H2 mass in kg} = 2.014 \times 1.66053906660 \times 10^{-27} \text{ kg} = 3.344325680132399 \times 10^{-27} \text{ kg} \][/tex]
### Step 3: Calculate the Reduced Masses
Reduced mass [tex]\(\mu\)[/tex] for the system is defined as:
[tex]\[ \mu = \frac{m_1 \times m_2}{m_1 + m_2} \][/tex]
When calculating I for different configurations:
#### For [tex]$SiH_4$[/tex] with Protium ([tex]$H_1$[/tex]):
[tex]\[ \mu_{SiH1} = \frac{4 \, (\text{Si mass} \times \text{H1 mass})}{(\text{Si mass} + \text{H1 mass})} \][/tex]
[tex]\[ I_{H1} = \mu_{SiH1} \times (\text{Bond length})^2 \][/tex]
Putting in the values:
[tex]\[ I_{H1} = 4 \times \left( \frac{ (4.6636239685461 \times 10^{-26}) \times (1.673557692882144 \times 10^{-27})}{4.6636239685461 \times 10^{-26} + 1.673557692882144 \times 10^{-27}} \right) \times (1.4798 \times 10^{-10})^2 = 1.4151257562138218 \times 10^{-46} \text{ kg m}^2 \][/tex]
#### For [tex]$SiH_4$[/tex] with Deuterium ([tex]$H_2$[/tex]):
[tex]\[ \mu_{SiH1} = \frac{3 \, (\text{Si mass} \times \text{H1 mass})}{(\text{Si mass} + \text{H1 mass})} + \frac{\text{Si mass} \times \text{H2 mass}}{\text{Si mass} + \text{H2 mass}} \][/tex]
[tex]\[ I_{H2} = \left( 3 \times \left( \frac{ (4.6636239685461 \times 10^{-26}) \times (1.673557692882144 \times 10^{-27})}{4.6636239685461 \times 10^{-26} + 1.673557692882144 \times 10^{-27}} \right) + \frac{ (4.6636239685461 \times 10^{-26}) \times (3.344325680132399 \times 10^{-27})}{4.6636239685461 \times 10^{-26} + 3.344325680132399 \times 10^{-27}} \right) \times (1.4798 \times 10^{-10})^2 = 1.7446845178774643 \times 10^{-46} \text{ kg m}^2 \][/tex]
### Step 4: Calculate the Change in Moment of Inertia
[tex]\[ \Delta I = I_{H2} - I_{H1} \][/tex]
[tex]\[ \Delta I = 1.7446845178774643 \times 10^{-46} \text{ kg m}^2 - 1.4151257562138218 \times 10^{-46} \text{ kg m}^2 = 3.2955876166364254 \times 10^{-47} \text{ kg m}^2 \][/tex]
### Summary of Results
- The bond length is [tex]\(1.4798 \times 10^{-10} \)[/tex] meters.
- The atomic masses of Si, Protium, and Deuterium are [tex]\(4.6636239685461 \times 10^{-26}\)[/tex] kg, [tex]\(1.673557692882144 \times 10^{-27}\)[/tex] kg, and [tex]\(3.344325680132399 \times 10^{-27}\)[/tex] kg respectively.
- The moment of inertia [tex]\(I_{H1}\)[/tex] for [tex]\(\text{SiH}_4\)[/tex] with protium atoms is [tex]\(1.4151257562138218 \times 10^{-46}\)[/tex] kg m[tex]\(^2\)[/tex].
- The moment of inertia [tex]\(I_{H2}\)[/tex] for [tex]\(\text{SiH}_4\)[/tex] when one hydrogen atom is replaced with deuterium is [tex]\(1.7446845178774643 \times 10^{-46}\)[/tex] kg m[tex]\(^2\)[/tex].
- The change in moment of inertia when deuterium replaces protium is [tex]\(3.2955876166364254 \times 10^{-47}\)[/tex] kg m[tex]\(^2\)[/tex].
