Let us say that we have one mole of $\\text{CO}_2$ gas at {temp}$^\\circ\\text{C}$ and 1 atm pressure.
", "advice": "For dilute gases we know $\\frac{n}{V}$ approaches zero. We can write vdW equation as $(P-\\frac{\\alpha n^2}{V^2}) V(1-\\frac{n}{V}b)=nRT$ setting the terms with $\\frac{n}{V}$ to zero we are left with the ideal gas equation $PV=nRT.
\n\nFor the volumes we have $P(V_2-nb)=nRT=PV_1$ which yields $V_2=V_1+nb$
", "rulesets": {}, "extensions": [], "variables": {"temp": {"name": "temp", "group": "Ungrouped variables", "definition": "random(18 .. 30#1)", "description": "", "templateType": "randrange"}, "vol_ideal": {"name": "vol_ideal", "group": "Ungrouped variables", "definition": "8.314*(temp+273.15)/(101325)", "description": "", "templateType": "anything"}, "vol_cor": {"name": "vol_cor", "group": "Ungrouped variables", "definition": "8.314*(temp+273.15)/(vol_ideal-4.27*10^-5) - (0.364^2/vol_ideal^2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["temp", "vol_ideal", "vol_cor"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "What volume does this gas occupy if you treat it as an ideal gas? Write your answer in $m^3$ using 4 significant figures.
", "minValue": "vol_ideal", "maxValue": "vol_ideal", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "The ideal gas law is an approximation. Realistically we need to consider both the size of the individual particles making the gas and their mutual attraction. If we take into consideration the attraction between gas molecules (for $\\text{CO}_2$ $a=0.364~\\frac{J\\cdot m^3}{mol^2}$) and the finite size of the molecules (for $\\text{CO}_2$ $b=4.27\\times10^{-5}~\\frac{m^3}{mol}$), what is the pressure (in Pa) of one mole of $\\text{CO}_2$ at {temp}$^\\circ\\text{C}$ occupying the volume you found in the previous question.
", "minValue": "vol_cor", "maxValue": "vol_cor+10", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "The van der Wall's equation is good at describing the relation between state variables for real gases. However in some special cases this equation can be simplified. Let us say that we have a very dilute gas (i.e $\\frac{n}{V}\\rightarrow 0$). What does the van der Wall's equation (assuming constant temperature and pressure) reduce to in these conditions? Write your answer as an equation relating state variables with $P$ and $V$ in one side and $n$,$R$ and $T$ on the other (i.e. $P^2V^3=nRT+T^2$).
", "answer": "PV=NRT", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "nrt", "value": ""}, {"name": "pv", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Let us consider only the correction to the ideal gas law that arises out of consideration for the volumes of the constituents particles. The ideal gas law (with these corrections) becomes $P(V-nb)=nRT$.
\nShow that for the same $P,n \\ \\text{and} \\ T$ the volume calculated from the ideal gas law with these corrections will be larger than if we were to use the ideal gas law $PV=nRT$ .
\nUse $V_2$ (written in numbas as ' V_2 ') to refer to the volume calculated using the ideal gas law with corrections and $V_1$ (written in numbas as ' V_1 ') as the volume calculated from the ideal gas law. Present your answer with $V_2$ written as a function of $V_1$, $V_2=f(V_1)$ (i.e. $V_2 = V_1+RT$).
", "answer": "V_2=V_1+nb", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "nb", "value": ""}, {"name": "v_1", "value": ""}, {"name": "v_2", "value": ""}]}], "contributors": [{"name": "Tiago Marinheiro", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4619/"}]}]}], "contributors": [{"name": "Tiago Marinheiro", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4619/"}]}