Activation Energy

The graph provided is a plot of versus , which is typically used to study the Arrhenius equation. According to the Arrhenius equation: Taking the natural logarithm of both sides, we get: This equation represents a straight line, where the slope is . From the graph: - The steepest slope corresponds to the highest activation energy , as the slope is proportional to . - The second steepest slope corresponds to , and - The least steep slope corresponds to . Therefore, based on the graph: Thus, the correct order of activation energies is . Therefore, the correct answer is (1) .

Step 1: Understanding paramagnetism.
Paramagnetic ions have unpaired electrons in their d- or f-orbitals. If all the electrons are paired, the ion is diamagnetic. We need to determine how many of the given ions have unpaired electrons.
Step 2: Electron configuration of each ion.
1. Mn :
Atomic number of Mn = 25. The electron configuration of Mn is . For Mn , the electron configuration becomes . Since there are 5 unpaired electrons, Mn is paramagnetic.
2. Cu :
Atomic number of Cu = 29. The electron configuration of Cu is . For Cu , the electron configuration becomes . Since there is 1 unpaired electron, Cu is paramagnetic.
3. Zn :
Atomic number of Zn = 30. The electron configuration of Zn is . For Zn , the electron configuration becomes . Since all electrons are paired, Zn is diamagnetic.
4. Yb :
Atomic number of Yb = 70. The electron configuration of Yb is . For Yb , the electron configuration becomes . Since there is 1 unpaired electron, Yb is paramagnetic.
5. Sc :
Atomic number of Sc = 21. The electron configuration of Sc is . For Sc , the electron configuration becomes . Since there are no unpaired electrons, Sc is diamagnetic.
6. La :
Atomic number of La = 57. The electron configuration of La is . For La , the electron configuration becomes . Since there are no unpaired electrons, La is diamagnetic.
7. Gd :
Atomic number of Gd = 64. The electron configuration of Gd is . For Gd , the electron configuration becomes . Since there are 7 unpaired electrons, Gd is paramagnetic.
8. Lu :
Atomic number of Lu = 71. The electron configuration of Lu is . For Lu , the electron configuration becomes . Since there is 1 unpaired electron, Lu is paramagnetic.
9. Ti :
Atomic number of Ti = 22. The electron configuration of Ti is . For Ti , the electron configuration becomes . Since there are no unpaired electrons, Ti is diamagnetic.
10. Ce :
Atomic number of Ce = 58. The electron configuration of Ce is . For Ce , the electron configuration becomes . Since there are no unpaired electrons, Ce is diamagnetic.
Step 3: Count the number of paramagnetic ions.
The paramagnetic ions are:
- Mn
- Cu
- Yb
- Gd
- Lu
Thus, the number of paramagnetic ions is 5.
Answer: \boxed{5}

Concept: Arrhenius equation:
Step 1:Substitute values}
Step 2:Take logarithm} Thus
Step 3:Solve}
Step 4:Convert to Celsius}

Step 1: Write Arrhenius Equations
Step 2: Calculate Ratio
Taking natural log: Step 3: Substitute Values
Given . K, . Final Answer: Nearest integer is 8.