diff --git a/notebooks/oscillator.jl b/notebooks/oscillator.jl
index 725145d746c179d95ec34d2f085eab1c7772feb5..59e9acb07a657e6a98b0b23d0f826386d33b06a3 100644
--- a/notebooks/oscillator.jl
+++ b/notebooks/oscillator.jl
@@ -165,9 +165,6 @@ function step_euler(y_n, A, Δt, α)
return lhs\rhs
end
-# ╔═╡ ce696808-9964-4741-ac4d-57f39ecb1f63
-system_matrix(δ) = [0. 1.; -1. -2. *δ]
-
# ╔═╡ 3086ab6c-fc5f-40e9-9700-130588f3f862
begin
fwd_euler(y_n, A, Δt) = step_euler(y_n, A, Δt, 0.)
@@ -175,6 +172,9 @@ begin
crank_nichols(y_n, A, Δt) = step_euler(y_n, A, Δt, 0.5)
end;
+# ╔═╡ ce696808-9964-4741-ac4d-57f39ecb1f63
+system_matrix(δ) = [0. 1.; -1. -2. *δ]
+
# ╔═╡ b1c1ae20-11a7-4451-bd3b-d20ce9d4965e
@bind δ0_exp Slider(collect(-4:0.05:1)[2:end], default=0.0, show_value=true)
@@ -271,9 +271,9 @@ version = "0.7.2"
[[deps.ColorSchemes]]
deps = ["ColorTypes", "ColorVectorSpace", "Colors", "FixedPointNumbers", "PrecompileTools", "Random"]
-git-tree-sha1 = "dd3000d954d483c1aad05fe1eb9e6a715c97013e"
+git-tree-sha1 = "be6ab11021cd29f0344d5c4357b163af05a48cba"
uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
-version = "3.22.0"
+version = "3.21.0"
[[deps.ColorTypes]]
deps = ["FixedPointNumbers", "Random"]
@@ -282,16 +282,10 @@ uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
version = "0.11.4"
[[deps.ColorVectorSpace]]
-deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "Requires", "Statistics", "TensorCore"]
-git-tree-sha1 = "a1f44953f2382ebb937d60dafbe2deea4bd23249"
+deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "SpecialFunctions", "Statistics", "TensorCore"]
+git-tree-sha1 = "600cc5508d66b78aae350f7accdb58763ac18589"
uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
-version = "0.10.0"
-
- [deps.ColorVectorSpace.extensions]
- SpecialFunctionsExt = "SpecialFunctions"
-
- [deps.ColorVectorSpace.weakdeps]
- SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
+version = "0.9.10"
[[deps.Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
@@ -742,6 +736,12 @@ git-tree-sha1 = "1aa4b74f80b01c6bc2b89992b861b5f210e665b5"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.21+0"
+[[deps.OpenSpecFun_jll]]
+deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
+uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
+version = "0.5.5+0"
+
[[deps.Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
@@ -749,9 +749,9 @@ uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"
[[deps.OrderedCollections]]
-git-tree-sha1 = "2e73fe17cac3c62ad1aebe70d44c963c3cfdc3e3"
+git-tree-sha1 = "d321bf2de576bf25ec4d3e4360faca399afca282"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
-version = "1.6.2"
+version = "1.6.0"
[[deps.PCRE2_jll]]
deps = ["Artifacts", "Libdl"]
@@ -914,6 +914,18 @@ version = "1.1.1"
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
+[[deps.SpecialFunctions]]
+deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
+git-tree-sha1 = "7beb031cf8145577fbccacd94b8a8f4ce78428d3"
+uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
+version = "2.3.0"
+
+ [deps.SpecialFunctions.extensions]
+ SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"
+
+ [deps.SpecialFunctions.weakdeps]
+ ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+
[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
diff --git a/notebooks/thermo_exercises.jl b/notebooks/thermo_exercises.jl
index fe88cbf7551abb65246f0621e693cb153cdf1ed3..ea22c06736f1104b7eeec2607b42dd467dbfb2d3 100644
--- a/notebooks/thermo_exercises.jl
+++ b/notebooks/thermo_exercises.jl
@@ -80,9 +80,41 @@ Derive expressions for entropy, internal energy, pressure, chemical potential, b
# ╔═╡ cf62429e-2d29-4ebe-a950-4f8a6c8ecc96
md"""
-From ``H = A = U - TS``,
+From Boltzmann's study of ideal gas, if N is large enough, we know that the internal energy of a monoatomic gas is purely kinetic energy. Hence, with each degree of freedom of motion for the gas particles contributing ``\frac{1}{2}\kboltz T`` to the internal energy, we have
+```math
+ U = \frac{3}{2}N\kboltz T
+```
+"""
+
+# ╔═╡ 86776c0e-67c9-4a78-a939-52cbfccbf59d
+function internal_energy(N, T)
+ return 1.5*N*BoltzmannConstant*T
+end
+
+# ╔═╡ 5492cb3d-002d-4bad-8ad6-072751eb1bf5
+md"""
+From the definition of ``F = U - TS``, we can derive
+```math
+\begin{align}
+S &= \frac{U-F}{T}\\
+S &= \frac{1}{T}\left(\frac{3}{2}N\kboltz T-N\kboltz T\left(\frac{\ln N \Lambda^3}{V}-1\right) \right)\\
+2S &= 3N\kboltz - 2N\kboltz\frac{\ln N \lambda^3}{V} + 2N\kboltz\\
+S &= \frac{1}{2}N\kboltz\left(5 - 2\frac{\ln N \Lambda^3}{V}\right)
+\end{align}
+```
"""
+# ╔═╡ b7d31193-8472-4764-ad7c-921e2d77e3ad
+function entropy(T, V, N, m)
+ Λ = debroglie_wavelength(T, m)
+ q1 = 0.5*N*BoltzmannConstant
+ q2 = 5-2*log(N)*Λ^3/V
+ return q1*q2
+end
+
+# ╔═╡ badc5384-2099-421e-aecf-8786f76cd422
+entropy(273.15u"K", 1u"L", 1*_mol, Ar.atomic_mass)
+
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
@@ -108,7 +140,7 @@ Unitful = "~1.15.0"
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
-julia_version = "1.9.0"
+julia_version = "1.9.2"
manifest_format = "2.0"
project_hash = "b586531e304952de66218a8af77aea24d18bbe87"
@@ -211,7 +243,7 @@ weakdeps = ["Dates", "LinearAlgebra"]
[[deps.CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
-version = "1.0.2+0"
+version = "1.0.5+0"
[[deps.ConcurrentUtilities]]
deps = ["Serialization", "Sockets"]
@@ -713,7 +745,7 @@ version = "0.42.2+0"
[[deps.Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
-version = "1.9.0"
+version = "1.9.2"
[[deps.PlotThemes]]
deps = ["PlotUtils", "Statistics"]
@@ -1166,7 +1198,7 @@ version = "0.15.1+0"
[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
-version = "5.7.0+0"
+version = "5.8.0+0"
[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
@@ -1228,6 +1260,10 @@ version = "1.4.1+0"
# ╠═9a8ac776-b1d3-4fb8-a360-5b341f86a2a7
# ╠═213dfc26-bfa2-406f-9fa3-1d168068e670
# ╟─5f7a9c35-97c3-4d4f-8bc4-494d43b65b94
-# ╠═cf62429e-2d29-4ebe-a950-4f8a6c8ecc96
+# ╟─cf62429e-2d29-4ebe-a950-4f8a6c8ecc96
+# ╠═86776c0e-67c9-4a78-a939-52cbfccbf59d
+# ╟─5492cb3d-002d-4bad-8ad6-072751eb1bf5
+# ╠═b7d31193-8472-4764-ad7c-921e2d77e3ad
+# ╠═badc5384-2099-421e-aecf-8786f76cd422
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002