# Math Help - Heat energy produced

1. ## Heat energy produced

The burning of 1.0 kg coal releases about 108 J of heat energy. How much energy could be produced if 1.0 kg of uranium was completely converted to energy?

How would I even solve for this? Do I need to find out something specific about the substance in question, such as its specific heat capacity?

2. I think that an error has been made in the writing of this question. I found the same question in my textbook and instead of it saying " $108J$" it says $10^8J$. I'm assuming the equation $E = mc^2$ must be used to figure out the answer (due to the large number)?

3. Originally Posted by RogueDemon
The burning of 1.0 kg coal releases about 108 J of heat energy. How much energy could be produced if 1.0 kg of uranium was completely converted to energy?

How would I even solve for this? Do I need to find out something specific about the substance in question, such as its specific heat capacity?
The key in this question is completely converted. To me this implies that it is talking about complete annihilation to form energy (such as matter-antimatter interaction).

Therefore you should use $E=mc^2$. You can also assume that $v << c$ hence relativity not applying.

The answer should be $c^2 = 9 \times 10^{16} \text { J}$

4. Thanks for the help. Just got one more question though. The instructions say to not use scientific notation and to use two decimal places when necessary. In that case, would the answer be $89,875,517,873,681,764J$? And how does the burning of 1.0kg of coal release only $10^8J$ of heat energy? Shouldn't the coal release the same amount of energy that the uranium releases?

5. Originally Posted by RogueDemon
Thanks for the help. Just got one more question though. The instructions say to not use scientific notation and to use two decimal places when necessary. In that case, would the answer be $89,875,517,873,681,764J$? And how does the burning of 1.0kg of coal release only $10^8J$ of heat energy? Shouldn't the coal release the same amount of energy that the uranium releases?
Use whatever the value of $c^2$ is, as it's a well recognised constant I see no reason why $c^2$ is wrong.

Coal releases comparatively little energy because it burns in a chemical reaction: $\text{C}{\text{(s)}} + \text{O}_2{\text{(g)}} \longrightarrow \text{CO}_2{\text{(g)}}$

From the equation we see that the $\text{O=O}$ bond is broken and two $\text{C=O}$ bonds formed. As bond breaking always releases energy and bond forming always absorbs energy the heat evolved is the difference. Each bond has a specific value of formation (for breaking it's the same value but different sign).

Uranium loses energy by radiation but in this question you're told to assume it's all converted to energy. Normally it loses energy by nuclear fission - the sum of the daughter nuclei is slightly less than the mass of uranium, this is called the mass defect and is calculated using $E=mc^2$. You've been told that it is all converted to energy, presumably by radiation rather than matter-antimatter annihilation.