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A random heating question, that randomly picks a material, and then heats it through either one or two phase changes, provides an example graph of the heating with scaled temperature ranges (though not with scaled latent and specific heats), and a table with the suitable constants.

a)

\n

$Q = m L_H$

\n

$Q = \\var{mass} \\times \\var{phase1_latentheat} = \\var{fusionheat}$

\n

\n

So the energy needed to melt the cube is {sigformat(fh_kj,3)} kJ.

\n

\n

b)

\n

$P_{net} = e \\sigma A \\Delta T^4$

\n

$P_{net} = e \\sigma A (T_1^4 - T_2^4)$

\n

where A is the surface area of the cube: $A = 6x^2 = 6 \\times (\\var{xx}^2) = 6*\\var{xx^2} = \\var{area}$

\n

$P_{net} = \\var{emissivity} \\times \\var{ksb2}\\times10^{-8} \\times \\var{area} \\times (\\var{phase1_temp}^4 - \\var{oven_temp}^4) = \\var{-power}$

\n

\n

So the net power emitted by the cube is {sigformat(-p_kw,3)} W , and the net power absorbed by the block is {sigformat(p_kw,3)} W.

\n

\n

c)

\n

$t = \\frac{E}{P}$

\n

$t = \\frac{\\var{fusionheat}}{\\var{sigformat(power,5)}} = \\var{time}$

\n

\n

So the time the cube takes to melt is {sigformat(time,3)} seconds.

\n

\n

d)

\n

$\\lambda_{max} = \\frac{K_W}{T}$

\n

$\\lambda_{max} = \\frac{\\var{kw}}{\\var{phase1_temp}} = \\var{wave}$

\n

\n

So the wavelength at which the cube's radiation is at a maximum is {sigformat(wave,3)} microns.

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g/cm3

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Calculate the total heat absorbed by the {Mat_string} as it melts in kJ, to three significant figures.

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Calculate the net power absorbed by the {mat_string} from radiation in kW, to three significant figures.

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Using the values from part a) and b), calculate how long the block took to melt in seconds, to three significant figures

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At what wavelength does the cube shine most brightly as it melts, in microns, to three significant figures?

", "variableReplacementStrategy": "originalfirst"}], "extensions": ["jsxgraph"], "statement": "

\n

A {mass_string} cube of solid {Mat_string} measures {x} cm along each side. It has a temperature of {phase1_temp} K, the fusion temperature of {mat_string}.  The cube sits in an oven with a temperature of {oven_temp} K and is heated only by radiation (no conduction or convection) until it entirely melts.

\n

Assume Stefan's constant $\\sigma = \\var{ksb2}\\times10^{-8}$  $W m^{-2}K^{-4}$ and Wien's displacement constant $k_W = \\var{kw}$ $\\mu mK$.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 Latent Heat of fusion emissivity {Mat_string} {phase1_latentheat} Jkg-1 {emissivity}
", "type": "question", "ungrouped_variables": ["switcharoo", "answer", "question_string"], "rulesets": {}, "contributors": [{"name": "Tom Stallard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/841/"}, {"name": "Josh Lim", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2990/"}]}]}], "contributors": [{"name": "Tom Stallard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/841/"}, {"name": "Josh Lim", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2990/"}]}