Exercice 2

4.1 μs

8.314

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begin
    n = 1.0
    P₁ = 1.0e5
    T₀ = 300.0
    S = 10e-4
    m = 10.0
    g = 9.81
    R = 8.314
end

104 ns

L'expression littérale de $x \equiv P_{2} / P_{1}$ vaut

$x = 1 + \frac{m g}{P_{1} S} .$

2.1 μs

1.981

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x = 1 + m * g / (P₁ * S)

91.0 ns

Bilan énergétique.

$Δ U = 0,$

$W = n R T_{0} (x - 1)$

et enfin

$Q = n R T_{0} (1 - x) .$

5.2 μs

2446.81

-2446.81

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begin
    ΔU = 0.0
    W = n * R * T₀ * (x - 1)
    Q = -W
    W, Q
end

112 ns

On a donc

$Δ S = - n R \log (x),$

$S^{r} = n R (1 - x)$

d'où on déduit

$S^{c} = n R [x - 1 - \log (x)] .$

3.4 μs

2.472568903205736

 
begin
    ΔS = -n * R * log(x)
    Sʳ = n * R * (1 - x)
    Sᶜ = n * R * (x - 1 - log(x))
end

311 ns

Aller plus loin...

7.1 μs

x
 
using Plots

382 μs

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default(lw = 2)

36.6 ms

visualize (generic function with 2 methods)

x
 
function visualize(step = 0.01)
    xlim, ylim = (0.01, 4.0), (-2, 2)
​
    X = range(xlim...; step)
​
    fig = plot(; xlim, ylim)
    plot!(fig, X, X .- 1, fill = (log.(X), 0.25, :blue), label = "x ↦ x - 1")
    plot!(fig, log, label = "ln")
end

43.8 μs

x
 
visualize()

2.2 ms