Problem Statement
Solve the work-energy problem: Find the temperature at which the average kinetic energy of translation of an ideal gas molecule equals $\bar{\varepsilon} = 0.040\ \text{eV}$. The average translational kinetic energy per molecule: $$\bar{\varepsilon} = \frac{3}{2}k_BT \implies T = \frac{2\bar{\varepsilon}}{3k_B}$$ Converting: $\ba
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Thermodynamics governs energy transformations involving heat and work. The First Law $\Delta U = Q – W$ expresses energy conservation. For an ideal gas, internal energy depends only on temperature ($U = nC_VT$), and the equation of state $PV = nRT$ links pressure, volume, and temperature.
- $\Delta U = Q – W$ — First Law of Thermodynamics
- $PV = nRT$ — ideal gas equation
- $C_P – C_V = R$, $\gamma = C_P/C_V$
- $W = \int P\,dV$ — work done by gas
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\bar{\varepsilon} = \frac{3}{2}k_BT \implies T = \frac{2\bar{\varepsilon}}{3k_B}$$
Answer
$$\boxed{v_f = \sqrt{2g h}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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