# Help regarding discovery question in Thermodynamics

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 =).

(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.)
• 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! Jan 22, 2022 at 11:41