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If the transfer of $1~\mathrm{kJ}$ of energy to a quantity of water causes its temperature to rise $10~^\circ\mathrm{C}$, what is the mass of the water? I know how to do the math and everything but I'm so confused with significant figures. If $1~\mathrm{kJ}$ or $100~\mathrm{J}$ only has 1 significant figure, then would I write $\pu{23.90 g}$ as $\pu{20 g}$?

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You have a reasonable question: if the data supplied specified 1.00 $\mathrm{kJ}$, and the temperature change was given as $10.00~^{\circ}\mathrm{C}$, then "23.90 g" would be a justifiable answer.

However, since the data precision was given to only one digit for energy, then the mass would be "20 g".

Personally, given enough space, I'd write an explanation, of the sort you've give, e.g. "Though I calculated 23.90 g, it could be rounded to 20 g due to lack of significant digits in the data given."

BTW, you've shown good effort in thinking through the homework assignment, in my opinion, which is as important as a numerically correct answer.

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