Question 11

I'm setting up the heat transfer between the two gas chambers through the partition, with side 1 at constant volume (Cv=3R/2) and side 2 at constant pressure (Cp=5R/2), then deriving the rate equations for how each temperature changes based on the conductive heat flow KAΔT/x. Combining the two rate equations gives dΔT/dt = -(16/15)(KA/xR)ΔT, so the temperature difference decays exponentially with time constant 15xR/(16KA). Solving for the halving time gives t = ln2 · (15/16) · xR/(KA), which numerically comes out to about 0.66 in the given units, and this result doesn't depend on which side starts hotter. **Setting up:** - S₁: rigid section → constant volume, heat capacity C_v = 3R/2 - S₂: piston keeps pressure constant → heat capacity C_p = 5R/2 - Heat current through partition: Q̇ = KA·ΔT/x, where ΔT = T₁ − T₂ **Rate equations:** dT₁/dt = −Q̇/C_v = −(2KA/3Rx)ΔT dT₂/dt = +Q̇/C_p = +(2KA/5Rx)ΔT d(ΔT)/dt = −(KA/xR)·(2/3 + 2/5)·ΔT = −(16/15)(KA/xR)·ΔT **Solving:** ΔT = ΔT₀ e^(−16KAt/15xR) For ΔT = ΔT₀/2: t = (15/16)(xR/KA)·ln2 = (15/16)(0.7)(xR/KA) ≈ 0.656 xR/KA So n ≈ 0.66

**Calculating numerical approximation** I need to solve something straightforward. The value I’m working with is n = 15/16 ln2, which equals 0.65625, approximate to .7. I’ll check my calculations involving the parameters. Half the time calculation leads to n .65625, confirmed as approximately 0.65 or 0.66 when I compare with official values. The user seems to expect a final JSON answer, so I’ll present the result as .65625, without rounding since it’s specified.

**Ensuring clarity in analysis** I need to confirm there are no hidden complications in my setup. Each section contains one mole of a monatomic gas. The partition, P1, has a likely negligible heat capacity and conducts heat but doesn't move. I'm considering Fourier's law for heat conduction. However, if it has a heat capacity, transient conduction becomes complex. I also need to track that S2 remains at atmospheric pressure, which influences calculations. Finally, I must analyze whether initial temperatures affect the magnitude of the difference. **Deciding on numerical value** I’m considering an alternative derivation for the effective heat capacity, which I calculated as C_eff = C1 C2 / (C1 + C2). That gives me a value of 15R/16. Using this, I find the time equation as n x R/KA, leading to n being about 0.65625 after approximating ln2 as 0.7. The expected numerical answer might be around 0.66. I’m pondering whether I should present it as the precise value or round it for clarity, given there’s no tolerance indicated.

Since P1 is immovable, S1 is at constant volume so heat added there follows dQ1 = nCv dT1, while S2's freely moving piston keeps it at constant pressure, giving dQ2 = nCp dT2. Heat transfers between the two sides through the partition connecting them. Setting up the conduction equation KA(T1-T2)/x for heat flowing through P1, I equate this to the rate of heat loss from S1 (via Cv) and heat gain by S2 (via Cp), using n=1 mole for simplicity. Defining ΔT = T1-T2, I combine the two rate equations to get a differential equation for how ΔT decays over time, involving the term K times a combination of 1/Cv and 1/Cp. For a monoatomic gas, plugging in Cv = 3R/2 and Cp = 5R/2 gives 1/Cv+1/Cp = 16/(15R), so ΔT decays exponentially as ΔT₀exp(-16KAt/(15Rx)). Setting ΔT = ΔT₀/2 and solving for t gives t = 15Rx ln2/(16KA), and substituting ln2≈0.7 yields t ≈ 0.656 xR/(KA), so the coefficient n comes out to about 0.66. Let me verify this by checking energy balance directly. Fourier's law says the conduction rate KAΔT/x depends only on temperature difference across the plate, not on volume changes, since S1 is rigid with constant volume, so its energy change comes purely from heat transfer with no work done, giving n Cv dT1 as the heat lost to S2. For S2, since it's insulated except through the conducting plate and expands at constant atmospheric pressure, its heat gain shows up as n Cp dT2, matching the conduction equation KA(T1-T2)/x on both sides — this confirms my earlier setup, and plugging in ln2=0.7 gives n ≈ 0.656. Looking at this problem, I need to analyze heat conduction between two gas sections with different constraint conditions. ## Setting Up the Physics **Section S1**: Bounded by fixed container wall and immovable partition P1 → constant volume process **Section S2**: Bounded by P1 and movable piston P2 at atmospheric pressure → constant pressure process **Heat conduction through P1** (Fourier's law, independent of volume changes): $$\frac{dQ}{dt} = \frac{KA(T_1-T_2)}{x}$$ **Energy balance for S1** (constant volume, only loses heat to S2): $$C_v\frac{dT_1}{dt} = -\frac{KA(T_1-T_2)}{x}$$ **Energy balance for S2** (constant pressure, receives heat, does work on piston): $$C_p\frac{dT_2}{dt} = \frac{KA(T_1-T_2)}{x}$$ ## Combining Equations Let $\Delta T = T_1 - T_2$. Subtracting the equations: $$\frac{d\Delta T}{dt} = \frac{dT_1}{dt} - \frac{dT_2}{dt} = -\frac{KA\Delta T}{x}\left(\frac{1}{C_v}+\frac{1}{C_p}\right)$$ For monoatomic gas: $C_v = \frac{3R}{2}$, $C_p = \frac{5R}{2}$ $$\frac{1}{C_v}+\frac{1}{C_p} = \frac{2}{3R}+\frac{2}{5R} = \frac{10+6}{15R} = \frac{16}{15R}$$ So: $$\frac{d\Delta T}{dt} = -\frac{16KA}{15Rx}\Delta T$$ ## Solving the Differential Equation $$\Delta T(t) = \Delta T_0 \exp\left(-\frac{16KA}{15Rx}t\right)$$ Setting $\Delta T = \Delta T_0/2$: $$\ln 2 = \frac{16KA}{15Rx}t$$ $$t = \frac{15Rx\ln 2}{16KA}$$ ## Calculating n Comparing with $t = \frac{nxR}{KA}$: $$n = \frac{15\ln 2}{16} = \frac{15 \times 0.7}{16} = \frac{10.5}{16} = 0.65625$$ ```json