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GR9277 #14
Problem
 GREPhysics.NET Official Solution Alternate Solutions
\prob{14}
The total energy of a blackbody radiation source is collected for one minute and used to heat water. The temperature of the water increases from 20.0 degrees Celsius to 20.5 degrees Celsius. If the absolute temperature of the blackbody were doubled and the experiment repeated, which of the following statements would be most nearly correct?

1. The temperature of the water would increase from 20 degrees Celsius to a final temperature of 21 degrees Celsius.
2. The temperature of the water would increase from 20 degrees Celsius to a final temperature of 24 degrees Celsius.
3. The temperature of the water would increase from 20 degrees Celsius to a final temperature of 28 degrees Celsius.
4. The temperature of the water would increase from 20 degrees Celsius to a final temperature of 36 degrees Celsius.
5. The water would boil within the one-minute time period

Statistical Mechanics$\Rightarrow$}Blackbody Radiation Formula

Recall

$P=ut\propto T^4,
$

where $P$ is the power and $u$ the energy and $T$ the temperature.

So, initially, the blackbody radiation emits $P_1=kT^4$. When its temperature is doubled, it emits $P_2=k(2T)^4=16kT^4$.

Recall that water heats according to $Q=mc\Delta T= \kappa \Delta T$. So, initially, the heat gain in the water is $Q_1=\kappa (0.5^\circ)$. Finally, $Q_2=\kappa x$, where $x$ is the unknown change in temperature.

Conservation of energy in each step requires that $kT^4t=\kappa/2$ and $16kT^4t=\kappa x$, i.e., that $P_i t = Q_i$. Divide the two to get $\frac{1}{16}=\frac{2}{x}\Rightarrow x=\Delta T = 8^\circ$. Assuming the experiment is repeated from the same initial temperature, this would bring the initial $20^\circ$ to $28^\circ$, as in choice (C).

Alternate Solutions
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mpdude8
2012-04-19 18:15:51
Whenever I see the word "blackbody", I think T^4. It seems like they always ask a question to see if you know the correct exponent in the blackbody-temperature relationship.
istezamer
2009-11-06 07:32:28
Of course we must start by knowing the fundamental equation that the Energy is proportional to Temperature^4.. so a double in temperature would increase the energy 16 folds!!
Now if we take the initial energy to be one unit...
1 unit increase the temperature 0.5 degrees
16 unit would increase the temperature 16 times 0.5 which is 8 degrees!! SO the correct choice is (C).
 pam d2011-09-28 09:56:39 Careful with the terminology, you should say that "radiative power" is proportional to temperature raised to the fourth power. Since the amount of time the experiment is run does not change, we have in this specific instance that the amount of energy transferred is proportional to temperature to the fourth power.
spacebabe47
2006-10-31 19:38:13
Dividing the two equations will actually get

1/16=.5/x

1/16=1/2x

x=8
Andresito
2006-03-24 19:20:59
In the expression P = u*t

Power = energy/time

Thus,

P = u/t

and this is consistent with Q = P*t

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