Temperature and Kinetic Theory

Temperature is the number shared by every body in mutual thermal equilibrium, made measurable by choosing a property that changes with hotness and fixing its scale at reproducible states; the kelvin takes that property to be the pressure of a dilute gas at constant volume. Use the supplied topic model Y_out = b*(1+t/273.15), where Y_out is p_B AT t (kPa). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Bath temperature t, measured in °C. Keep a (Sample A pressure at 0 °C) = 100 kPa; b (Sample B pressure at 0 °C) = 60 kPa fixed. For this exercise use only -212.813 °C ≤ t ≤ 62.8125 °C on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 50.185338 kPa, with a bounded reading uncertainty ±3.5418 kPa. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0275625 °C, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The linear expansion coefficient α is the fractional change in length per kelvin, α = (1/L)(dL/dT), so a free member of initial length L₀ warmed by ΔT grows by ΔL ≈ αL₀ΔT. Use the supplied topic model Y_out = gap/(0.012*len), where Y_out is GAP CLOSES AT ΔT (K). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. len is Rod length L₀, measured in m. Keep dt (Temperature rise ΔT) = 60 K; gap (Clearance s) = 0.5 mm fixed. For this exercise use only 1.425 m ≤ len ≤ 4.575 m on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 12.450819 K, with a bounded reading uncertainty ±0.5015 K. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown len by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315 m, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The ideal-gas equation of state, pV = nRT, is the single relation among pressure, volume, absolute temperature and amount of substance that every sufficiently dilute gas obeys, with R = 8.314 J mol⁻¹ K⁻¹ the same for all of them. Use the supplied topic model Y_out = 60.22/v, where Y_out is NUMBER DENSITY N/V (×10²⁵ m⁻³). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. v is Volume V, measured in L. Keep t (Absolute temperature T) = 300 K fixed. For this exercise use only 7.125 L ≤ v ≤ 22.875 L on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 3.598984 ×10²⁵ m⁻³, with a bounded reading uncertainty ±0.14496 ×10²⁵ m⁻³. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown v by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.001575 L, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The kinetic model states that gas pressure is the momentum delivered to unit wall area per second by molecular collisions, which for point molecules in random motion comes to p = ⅓ n m ⟨v²⟩, with ⟨v²⟩ the mean of the squared speeds. Use the supplied topic model Y_out = exp(-0.5*(vp/sx)^2), where Y_out is COUNT AT PROBE ÷ AT vₓ = 0 (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. vp is Probe class vₓ, measured in m s⁻¹. Keep sx (Rms x-speed σₓ) = 300 m s⁻¹; n (Number density n) = 2.5 ×10²⁵ m⁻³ fixed. For this exercise use only 170 m s⁻¹ ≤ vp ≤ 1430 m s⁻¹ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.0074893179, with a bounded reading uncertainty ±0.0011217. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown vp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.126 m s⁻¹, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.