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new to the forum and I'm currently learning about "heat capacity" in my physics book, and I've had a bit of a tough time trying to see the reasoning behind a certain question. In this book, some concepts are left to be "realized" and I don't quite get what they're trying to convey.

The problem: You fill a tub with 3.6 litres of warm water with the temperature 45$^\circ$C. How much water with the temperature 13$^\circ$C should you add to the tub in order for the temperature of the tub to be 37$^\circ$C? Disregard energy conversion from the surrounding.

And then the solution they give boils down to this.

Mass of water in the tub = $m$ = 3.6kg . Let the specific heat capacity for water = $c$ Let the change in Temperature $\Delta$T = $T_A$ - $T_B$ where $T_A$ represents the temperature after and $T_B$ represents the temperature before within the tub. Let the mass of the water at 13$^\circ$C be m$_n$ .

$c * 3.6 * (37-45) J = c * m_n * (37-13) J $

And from here you can calculate the mass of water needed to cool down the bath tub. What I don't quite understand is why the change in the systems energy at the temperature which one would be looking for is equivalent to the change in energy between the temperature we're trying to reach and the change we're adding. If the question or anything I've written is unclear, please don't hesitate to ask for clarification, I will do my best. Thank you for taking the time to help me with this =).

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It's because of conservation of energy.

There are two masses of water involved: the water originally in the tub, and the water that you're adding. Because energy is conserved, the thermal energy that flows out of the water originally in the tub must be equal to the thermal energy that flows into the water you add. The equation you've written down just expresses that fact mathematically.

(As an aside, the equation you've written down isn't quite right; you'll get a negative answer if you calculate $m_n$ from it. See if you can figure out how to fix it, given the above facts.)

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  • $\begingroup$ So if I've understood this correctly: The energy difference within the system which would be needed to reach a certain temperature, is equivalent to the energy within the system when the system has that temperature afterwards, as well as the energy added by the new "things" (not sure what to call things you add into the system) into the system, and with the use of that you can mathemtically re arrange this to get the mass? I'm thinking the underlying fundamental theorem here is that energy within the system is preserved and so what's being compared here are the forms of the system. $\endgroup$
    – Erade
    Jan 20, 2022 at 15:10
  • $\begingroup$ @Erade: I'm not sure I follow your first sentence, but I agree with the last one. $\endgroup$ Jan 20, 2022 at 16:40
  • $\begingroup$ I was finally able to get a proper grasp of what's going on here, thanks for your help! Essentially, the energy gained by the mass of water within the system which had a temperature increase is equivalent to the energy lost by the mass of water which had a temperature decrease. This is because energy within a system cannot be lost, only converted, and since the question asks to disregard other factors, conversion here happens through the form of thermal energy, which eventually disperses through the body of water to an average temperature. $\endgroup$
    – Erade
    Jan 22, 2022 at 11:36
  • $\begingroup$ This is where $ cm_1*(T_n - T_l) = - cm_2*(T_n-T_h) $ comes in as the energy gain required for the new total temperature Tn of mass m1 is equivalent to the energy loss of the mass m2 with a new temperature of Tn . I might be wrong on all this again but I feel like it's gotten alot easier now, just wanted to say thanks for replying to me! $\endgroup$
    – Erade
    Jan 22, 2022 at 11:41
  • $\begingroup$ @Erade: No problem. If my answer helped solve your question, you can "accept" it as the best answer by clicking on the check-mark to the left of the answer text. $\endgroup$ Jan 22, 2022 at 14:18

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