The internal energy of a monoatomic gas is 3nRT$3nRT$. One mole of helium is kept in a cylinder having internal cross section area of 17 cm²$17 \, \mathrm{cm}^2$ and fitted with a light movable frictionless piston. The gas is heated slowly by suppling 126 J$126 \, \mathrm{J}$ heat. If the temperature rises by 4°C$4^{\circ}\mathrm{C}$ , then the piston will move ____ cm.
(atmospheric pressure = 10⁵ Pa$10^{5} \, \mathrm{Pa}$)
A.14.5$14.5$
B.1.55$1.55$
C.15.5$15.5$
D.1.45$1.45$
Solution & Explanation
Related Formula
Δ Q = Δ U + W$$\Delta Q = \Delta U + W$$Δ U = n Cv Δ T$$\Delta U = n C_v \Delta T$$W = P Δ V = P (A · Δ x)$$W = P \Delta V = P (A \cdot \Delta x)$$
Core Logic
For a monoatomic gas like Helium, Cv = (3)/(2)R$C_v = \frac{3}{2}R$. Given internal energy is 3nRT$3nRT$, but standard change is Δ U = n Cv Δ T$\Delta U = n C_v \Delta T$. Wait, the problem states internal energy U = 3nRT$U = 3nRT$ (which implies Cv = 3R$C_v = 3R$ for this specific state/setup, or it's a typo in the paper and they mean U = (3)/(2)nRT$U = \frac{3}{2}nRT$). Let's follow the solution's lead: Δ U = 3nRΔ T$\Delta U = 3nR\Delta T$ is directly given.
Step 1: Calculate Change in Internal Energy
Using the given relation:
Δ U = 3nR Δ T$$\Delta U = 3nR \Delta T$$
Substitute n=1$n=1$, R = (25)/(3) J/mol · K$R = \frac{25}{3} \, \mathrm{J/mol \cdot K}$, and Δ T = 4°C$\Delta T = 4^{\circ}\mathrm{C}$:
Work done is isobaric since the piston is freely moving:
W = P Δ V = P · A · Δ x$$W = P \Delta V = P \cdot A \cdot \Delta x$$26 = 10⁵ × 17 × 10⁻⁴ × Δ x$$26 = 10^{5} \times 17 \times 10^{-4} \times \Delta x$$26 = 170 · Δ x$$26 = 170 \cdot \Delta x$$Δ x = (26)/(170) m = (26)/(170) × 100 cm = 15.3 cm$$\Delta x = \frac{26}{170} \, \mathrm{m} = \frac{26}{170} \times 100 \, \mathrm{cm} = 15.3 \, \mathrm{cm}$$
Closest given option is 15.5 cm$15.5 \, \mathrm{cm}$.
Pattern Recognition
When dealing with a free movable piston, pressure is constant. Apply First Law directly. Watch out for modified U$U$ definitions given in the prompt, replacing standard Cv$C_v$ logic.
Practice past-year questions one chapter at a time. Pick an exam → subject → chapter and get every PYQ for that topic — pulled together from all past papers — with the chapter's key formulas alongside.
