To determine which of the given statements (P, Q, R, S) are false, we will analyze each statement step-by-step.
### Step 1: Analyze Statement P
**Statement P**: "Orbital angular momentum of the azimuthal quantum number as lowest for this principle quantum number is \( \frac{h}{\pi} \)."
- The formula for orbital angular momentum \( L \) is given by:
\[
L = \sqrt{l(l + 1)} \cdot \frac{h}{2\pi}
\]
where \( l \) is the azimuthal quantum number.
- The lowest value of \( l \) for any principal quantum number \( n \) is \( 0 \) (for s-orbitals).
- Therefore, the lowest orbital angular momentum is:
\[
L = \sqrt{0(0 + 1)} \cdot \frac{h}{2\pi} = 0
\]
- The statement claims that the lowest angular momentum is \( \frac{h}{\pi} \), which is incorrect.
**Conclusion for P**: **False**
### Step 2: Analyze Statement Q
**Statement Q**: "If \( n=3 \), \( l=0 \), \( m=0 \) for the last valence shell electron, then the possible atomic number may be 12 or 13."
- For \( n=3 \) and \( l=0 \), we are referring to the 3s subshell.
- The electron configuration leading to the 3s subshell can be:
- \( 1s^2 2s^2 2p^6 3s^2 \) (Atomic number 12, Magnesium)
- \( 1s^2 2s^2 2p^6 3s^1 \) (Atomic number 11, Sodium)
- The statement suggests atomic numbers 12 or 13, but 13 corresponds to Aluminum, which has a 3p electron, not a 3s electron.
**Conclusion for Q**: **False**
### Step 3: Analyze Statement R
**Statement R**: "Total spin of electrons for the atom \( _{25}Mn \) is \( \pm \frac{7}{2} \)."
- The electron configuration for \( Mn \) (atomic number 25) is \( [Ar] 3d^5 4s^2 \).
- The total number of unpaired electrons in \( 3d^5 \) is 5 (since all five d-orbitals are singly occupied).
- The total spin \( S \) is given by:
\[
S = \text{Number of unpaired electrons} \times \frac{1}{2} = 5 \times \frac{1}{2} = \frac{5}{2}
\]
- The total spin can be \( +\frac{5}{2} \) or \( -\frac{5}{2} \), not \( \pm \frac{7}{2} \).
**Conclusion for R**: **False**
### Step 4: Analyze Statement S
**Statement S**: "Spin magnetic moment of inert gas is zero."
- Inert gases (like Helium, Neon, Argon) have completely filled electron shells.
- A completely filled shell means there are no unpaired electrons, resulting in a net magnetic moment of zero.
**Conclusion for S**: **True**
### Final Conclusion
The false statements are **P, Q, and R**. The only true statement is **S**.
### Summary of False Statements:
- **P**: False
- **Q**: False
- **R**: False
- **S**: True
