Problem Statement
Find the minimum electron kinetic energy needed to ionize a helium atom from the ground state (first ionization energy = 24.6 eV).
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
Find the minimum electron kinetic energy needed to ionize a helium atom from the ground state (first ionization energy = 24.6 eV).
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Find the minimum electron kinetic energy needed to ionize a helium atom from the ground state (first ionization energy = 24.6 eV).
Solution
In an electron-atom collision, the bombarding electron must have enough kinetic energy to ionize the atom. Threshold condition (with the electron coming to rest after giving all energy):
$$T_{threshold} = E_i = 24.6 \text{ eV}$$
However, this exact threshold assumes the ejected electron has zero kinetic energy and the ion takes negligible recoil. In practice, the threshold is:
$$T_{min} = \frac{(m + M)E_i}{M} \approx E_i$$
since the electron mass $m \ll$ He mass $M$. So $T_{min} = 24.6$ eV.
Answer: $T_{min} = 24.6$ eV (first ionization energy of He)
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{T_{min} = \frac{(m + M)E_i}{M} \approx E_i}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{T_{min} = \frac{(m + M)E_i}{M} \approx E_i}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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