2d:null 2e:["$","title",null,{"children":"Hydrogen loading of Palladium | Matthew Lilley | Observable"}] 33:T5a6,md` This is an Observable notebook - a super convenient way to interact with and visualise data. If you want to know more about how these notebooks work, I suggest heading over to the [5 min into](https://observablehq.com/@observablehq/five-minute-introduction). Otherwise, please enjoy this interactive physics experience 🤓. The purpose of this notebook is to allow people to interactively explore the "phase space" of [palladium-hydride](https://en.wikipedia.org/wiki/Palladium_hydride) in order to help create intuition for how the palladium hydrogen system behaves at different temperatures. [Santandrea and Behrens](https://www.degruyter.com/downloadpdf/j/htmp.1986.7.2-3/htmp.1986.7.2-3.149/htmp.1986.7.2-3.149.pdf) have made an experimental review of the relationships between hydrogen gas pressure (P), temperature (T) and H/Pd ratio loading ratio (x) in the palladium hydrogen system under various conditions. *(note: we are using the more common notation 'x' to denote loading ratio as opposed to 'r' which Santandrea and Behrens use)* On pages 154 & 155, formulas are given that describe the "isotherms" for different phases of palladium-hydride: ${tex`\alpha`}, ${tex`\beta`} and ${tex`\alpha + \beta`}. These isotherms describe the relationship between desorption pressure of hydrogen and loading ratio for a fixed temperature. We will explore these in the rest of the notebook. *Units for this notebook: T in K, P in atm* `34:T72b,{ let target = DOM.element('div'); let T = TC+273.15 // calculating mixed alpha+beta phase pressure here so that it is not recalculated during the rendering of every point on the curve var mixedP = Math.exp(-4693.0/T + 10.97); // Show alpha, beta and alpha+beta as separate curves if (curveSelect.value=="separate"){ functionPlot({ grid: true, data: [ {fn: `exp( 12.9 - 2330/${T} - (x/${T})*(4530 + 2.017e6/${T}) )*(x^2/(1-x)^2)`, graphType: 'polyline', color:"#29B93E"}, {fn: `exp(12.9 - (1/${T})*(12070 - 10830*x))*(x^2/(1-x)^2)`, graphType: 'polyline', color:"#29AEB9"}, {fn:`exp(-4693.0/${T} + 10.97)`, graphType: 'polyline', color:"#B99729"}, ], xAxis: {domain: FullRange.loading,label:'Loading ratio'}, yAxis: {domain: FullRange.pressure,label:'Pressure (atm)'}, target:target }); } // Show the complete phase curve by stitching alpha, beta and alpha+beta together else { functionPlot({ grid: true, data: [ { graphType: 'polyline', color:"#2980B9", fn: function (scope) { var x = scope.x // x<0.1 has been chosen based on my obserations of the alpha only curve. // Specifically, the first intersection with the mixed curve always happens below x=0.1. if(x<0.1) { var alphaP = Math.exp( 12.9 - 2330/T - (x/T)*(4530 + 2.017e6/T) )* Math.pow(x/(1-x),2.0) return alphaP <= mixedP ? alphaP : mixedP } else { var betaP = Math.exp(12.9 - (1/T)*(12070 - 10830*x))*Math.pow(x/(1-x),2.0) return betaP >= mixedP ? betaP : mixedP } } } ], xAxis: {domain: FullRange.loading,label:'Loading ratio'}, yAxis: {domain: FullRange.pressure,label:'Pressure (atm)'}, target:target }); } return target; }35:[] 31:{"id":"2a0a148d03f541e2","slug":"hydrogen-loading-of-palladium","title":"Hydrogen loading of Palladium","thumbnail":"397406a18745ee1a37b066c0ec1678147749a3eef1f0cb6712341018fdbac408","access_level":"public","create_time":"$D2019-10-07T08:26:59.100Z","update_time":"$D2020-09-12T07:27:51.551Z","view_time":null,"first_public_time":"$D2019-10-07T12:15:02.310Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"4a28ea9ed826d0a0","login":"mklilley","name":"Matthew Lilley","avatar_url":"https://avatars-next.observableusercontent.com/5b842c5b0f775dd6c0697d886b371e2b0ef06d5a7fdd2064fec88b6200598484","tier":"starter_2024"},"version":828,"publish_version":828,"role":null,"authored":false,"authors":[{"type":"individual","id":"dcd27f799cb1f0fc","login":"mklilley","name":"Matthew Lilley","avatar_url":"https://avatars-next.observableusercontent.com/5b842c5b0f775dd6c0697d886b371e2b0ef06d5a7fdd2064fec88b6200598484","tier":null}],"members":[{"type":"individual","id":"dcd27f799cb1f0fc","login":"mklilley","name":"Matthew Lilley","avatar_url":"https://avatars-next.observableusercontent.com/5b842c5b0f775dd6c0697d886b371e2b0ef06d5a7fdd2064fec88b6200598484","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Hydrogen loading of Palladium","cells":[{"id":0,"value":"md`# Hydrogen loading of Palladium`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":152,"value":"md`## ${tex`\\alpha`} phase`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":155,"value":"md`When the loading ratio is low (${tex`x \\lesssim 0.017`} at room temperature), the Pd lattice expands only slightly as hydrogen is absorbed - this is called the ${tex`\\alpha`} phase. The formula describing the isotherms in this phase is given by:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":181,"value":"md`${tex.block`\\ln{P} = 12.9 + 2\\ln{\\frac{x}{1-x}} - \\frac{x}{T}\\left( \\frac{2330}{x} + 4530 + \\frac{2.017 \\times 10^6}{T}\\right)`}`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":119,"value":"md`## ${tex`\\beta`} phase`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":147,"value":"md`At room temperature, when the loading ratio rises above ${tex`x \\gtrsim 0.017`}, parts of the Pd lattice expand significantly. This expanded palladium is called ${tex`\\beta`} phase palladium. When the loading is sufficiently high (${tex`x \\gtrsim 0.58`}) the ${tex`\\alpha`} phase disappears completely and we are left with only ${tex`\\beta`} phase palladium. The formula describing the isotherms in this phase is given by:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"md`${tex.block`\\ln{P} = 12.9 + 2\\ln{\\frac{x}{1-x}} - \\frac{1}{T}\\left( 12070 - 10830 x \\right)`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":282,"value":"md`## ${tex`\\alpha + \\beta`} phase`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":291,"value":"md`For intermediate levels of loading, i.e ${tex` 0.017\\lesssim x \\lesssim 0.58`} at room temperature, the palladium lattice is in a mixed phase of ${tex`\\alpha + \\beta`}. In this phase, the desorption pressure only depends on the temperature and not the loading ratio and is given by:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":286,"value":"md`${tex.block`\\ln{P} = -4693.0/T + 10.97`}`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":785,"value":"md`## The phase diagram\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":330,"value":"md`\nThe boundaries between the different phases of palladium are not constant - they depend on temperature.\n\nChange the temperature below to see how it impacts the following phase diagram`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":311,"value":"md `T (C) = ${bind(html``, viewof TC)} ${bind(html``, viewof TC)} `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":752,"value":"md`*You can interact with the graph below - hover over and use the mouse pad or scroll wheel to zoom in and out. Below the graph are options to reset the axes using 'Initial ranges' and to switch view from \"combined\" to \"separate\" in order to see each phase curve separately. *`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":588,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":598,"value":"\nviewof FullRange = {\n const form = html``;\n form.oninput = () => form.value = {\n pressure: [form.minP.valueAsNumber,form.maxP.valueAsNumber],\n loading: [form.minR.valueAsNumber,form.maxR.valueAsNumber]\n };\n form.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":720,"value":"viewof curveSelect = form(html`\n`)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":374,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"md`### Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":452,"value":"md`\nInspiration for some of the code comes from\n\nhttps://observablehq.com/@mbostock/synchronized-views\nhttps://observablehq.com/@mbostock/inline-inputs\nhttps://observablehq.com/@mbostock/multi-value-inputs\n\nBelow you'll find code that's required to make the features of this notebook work.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":50,"value":"functionPlot = {\n window.d3 = await require('d3@3');\n return require('https://cdn.bootcss.com/function-plot/1.18.1/function-plot.js');\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":381,"value":"function bind(input, view) {\n input.value = view.value;\n input.oninput = () => view.value = input.valueAsNumber;\n\n function update({value}) {\n if (!isNaN(value) && input.valueAsNumber !== value) {\n input.value = value;\n }\n }\n\n view.addEventListener(\"input\", update);\n return Generators.disposable(input, () => {\n view.removeEventListener(\"input\", update);\n }), input;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":400,"value":"class View {\n constructor(value) {\n Object.defineProperties(this, {\n _list: {value: [], writable: true},\n _value: {value, writable: true}\n });\n }\n get value() {\n return this._value;\n }\n set value(value) {\n this._value = value;\n this.dispatchEvent({type: \"input\", value});\n }\n addEventListener(type, listener) {\n if (type != \"input\" || this._list.includes(listener)) return;\n this._list = [listener].concat(this._list);\n }\n removeEventListener(type, listener) {\n if (type != \"input\") return;\n this._list = this._list.filter(l => l !== listener);\n }\n dispatchEvent(event) {\n const p = Promise.resolve(event);\n this._list.forEach(l => p.then(l));\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":403,"value":"viewof TC = new View(20)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":704,"value":"import {form} from \"@mbostock/form-input\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 32:"$31:body"