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{"name": "Integration: Solving Separable Differential Equations - Radioactive Decay", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: Solving Separable Differential Equations - Radioactive Decay", "tags": [], "metadata": {"description": "<p>Solving a separable differential equation that describes the rate of decay of radioactive isotopes over time with a known initial condition to calculate the mass of the isotope after a given time and the time taken for the mass to reach $m$ grams.&nbsp;</p>\n<p><a href=\"https://www.nuclear-power.com/nuclear-power/reactor-physics/atomic-nuclear-physics/radioactive-decay/radioactive-decay-law/decay-constant/\">Decay Constant - Radioactivity - Nuclear Power (nuclear-power.com)</a></p>", "licence": "None specified"}, "statement": "<p>The rate of decay of the radioactive isotope $\\var{isotopes_names[j]}$ is given by:</p>\n<p>\\[\\frac{\\mathrm{d}M}{\\mathrm{d}t}=-\\var{lambda}M\\]</p>\n<p>where $M$ is the mass of the isotope in grams and $t$ is time measured in $\\var{timescale[j]}$.<br/>A sample initially contains $\\var{m0}$ grams of the isotope.</p>", "advice": "<p>a) To calculate the mass of the isotope at a specific time, we need an expression for how the mass, $M$, changes in terms of time, $t$. This is given by the differential equation,</p>\n<p>\\[\\frac{\\mathrm{d}M}{\\mathrm{d}t}=-\\var{lambda}M\\]</p>\n<p>which we can solve using separation of variables.</p>\n<p>First we need to separate the variables between the two sides:</p>\n<p>\\[\\frac{1}{M}dM=-\\var{lambda}dt\\]</p>\n<p>Now we can integrate both sides:&nbsp;</p>\n<p>\\[ \\begin{split} \\int\\frac{1}{M}dM&amp;=\\int -\\var{lambda}dt \\\\ \\ln{M}&amp;= -\\var{lambda}t + c \\end{split} \\]</p>\n<p>Taking the exponential of both sides:</p>\n<p>\\[ \\begin{split} M&amp;= e^{ -\\var{lambda}t + c} \\\\ M&amp;=e^c\\times e^{-\\var{lambda}t }\\end{split} \\]</p>\n<p>Now, $e^c$ is a constant which we can calculate using the initial contition that $m=\\var{m0}$ when $t=0$. By substituting in the values in the equation we get:</p>\n<p>\\[ \\begin{split}&nbsp; \\var{m0}&amp;=e^c\\times e^{-\\var{lambda}\\times 0 } \\\\ \\var{m0}&amp;=e^c\\times e^0 \\\\ \\var{m0}&amp;=e^c\\times 1 \\\\ \\var{m0}&amp;=e^c \\end{split} \\]</p>\n<p>So, $e^c=\\var{m0}$. Therefore,</p>\n<p>\\[ \\begin{split} M&amp;=\\var{m0} e^{-\\var{lambda}t }\\end{split} \\]</p>\n<p>We can now substitute $t=\\var{t1}$ to calculate the mass at that time:&nbsp;</p>\n<p>\\[ \\begin{split} M&amp;=\\var{m0} e^{-\\var{lambda} \\times \\var{t1} } \\\\ M&amp;= \\var{m0}\\times e^{\\simplify{{-decayconstant[j]*t1}}}\\\\ M&amp;= \\var{m0} \\times \\simplify{{e^(-decayconstant[j]*t1)}} = \\var{ansa} ~~\\text{[3 s.f.]} \\end{split} \\]</p>\n<p>b) To calculate how long it takes for the mass of the isotope in the sample to fall to $\\var{m1}$ gram, we will use the expression&nbsp;</p>\n<p>\\[ \\begin{split} M&amp;=\\var{m0} e^{-\\var{lambda}t }\\end{split} \\]</p>\n<p>found in part (a). We can substitute $M=\\var{m1}$ and solve the equation for t.</p>\n<p>\\[ \\begin{split} \\var{m1}&amp;=\\var{m0} e^{-\\var{lambda}t } \\\\ \\frac{\\var{m1}}{\\var{m0}}&amp;=e^{-\\var{lambda}t }\\end{split} \\]</p>\n<p>Taking $\\ln()$ of both sides:</p>\n<p>\\[ \\begin{split} \\ln \\left(\\frac{\\var{m1}}{\\var{m0}}\\right)&amp;=-\\var{lambda}t \\\\ -\\frac{\\ln \\left(\\frac{\\var{m1}}{\\var{m0}}\\right)}{\\var{lambda}}&amp;=t \\end{split} \\]</p>\n<p>Thus, $m=\\var{m1}$ when $t=\\var{ansb}$ {timescale[j]}.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"lambda": {"name": "lambda", "group": "Ungrouped variables", "definition": "scientificnumberlatex( decayconstant[j])", "description": "", "templateType": "anything", "can_override": false}, "m0": {"name": "m0", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "t1": {"name": "t1", "group": "Ungrouped variables", "definition": "random (10..500 #10)", "description": "", "templateType": "anything", "can_override": false}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "m0/2", "description": "", "templateType": "anything", "can_override": false}, "ansA": {"name": "ansA", "group": "Ungrouped variables", "definition": "siground(m0*exp(-decayconstant[j]*t1),3)", "description": "", "templateType": "anything", "can_override": false}, "ansB": {"name": "ansB", "group": "Ungrouped variables", "definition": "round (-ln(0.5)/decayconstant[j])", "description": "", "templateType": "anything", "can_override": false}, "timescale": {"name": "timescale", "group": "Ungrouped variables", "definition": "[ \"years\", \"years\", \"minutes\",\"seconds\"]", "description": "", "templateType": "anything", "can_override": false}, "isotopes_names": {"name": "isotopes_names", "group": "Ungrouped variables", "definition": "[\"Carbon 14\", \"Radium 226\", \"Free Neutron 239\", \"Radon 220\"]", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "DecayConstant": {"name": "DecayConstant", "group": "Ungrouped variables", "definition": "[ (1.24*10^(-4)) , 4.273*10^(-4) , 1.1*10^(-3) , 1.33*10^(-2) ]", "description": "", "templateType": "anything", "can_override": false}, "step1": {"name": "step1", "group": "Ungrouped variables", "definition": "-decayconstant[j]*t1", "description": "", "templateType": "anything", "can_override": false}, "step2": {"name": "step2", "group": "Ungrouped variables", "definition": "exp(step1)", "description": "", "templateType": "anything", "can_override": false}, "step3": {"name": "step3", "group": "Ungrouped variables", "definition": "m0*step2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "(ansA<m0)", "maxRuns": 100}, "ungrouped_variables": ["lambda", "m0", "t1", "m1", "ansA", "ansB", "timescale", "isotopes_names", "DecayConstant", "j", "step1", "step2", "step3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the mass of the isotope present in the sample after $\\var{t1}$ $\\var{timescale[j]}$?</p>\n<p>The mass is [[0]]grams.</p>\n<p><em>Round your answer to 3 significant figures, if needed.</em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, 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