When you take a bath, how many kilograms of hot water (50°C) must you mix with c

Question

When you take a bath, how many kilograms of hot water (50°C) must you mix with cold water (10°C) so that the temperature of the bath is36°C? The total mass of water (hot plus cold) is 187 kg. Ignore any heat flow between the water and itsexternal surroundings.

Explanation / Answer

STEP ONE: Gathering Information ---------- When you take a bath, how many kilograms of hotwater (50 oC) must you mix with cold water (10oC) so that the temperature of the bath is 36oC? From this we have: Thi = 50 oC   (initial temperatureof hot water) Tci = 10 oC   (initial temperatureof cold water) Tf = 36 oC     (finaltemperature of the system) ---------- The total mass of water (hot plus cold) is 188kg. From this we have: mt = 187kg STEP TWO: Setting up the equation Seeing that we can ignore heat flow between the water and itsexternal surroundings, we can say that: Qcold water = -Qhot water    Qcold water is positive because it is gainingenergy. -Qhot water is negative because it is losingenergy.     (Note: heat flows fromhot to cold objects) And we also know that Q = mcT (where m = mass, c =specific heat, T = Tf - Ti = change intemp) So now we can write out the full equation: (1)---------------- Qcold water = -Qhotwater    (2)---------------- mcwcwT =-mhwcwT   (3)---------------- mcwcw(Tf -Tci) = -mhwcw(Tf- Thi)    Now cw is the specific heat for water. Because itis the same value on both sides of the equation, we can cancel themout to get: (4)---------------- mcw(Tf -Tci) = -mhw(Tf -Thi) Now there is a problem as we do not have the values formcw, as mhw (mass of hot water) is waht weare trying to find. However we do know the total mass of thehot and cold water, so: mcw + mhw = 187 kg  ---------------> mcw = 187 kg - mhw Substituting this value in we have our final equation: (5)---------------- (187 kg - mhw)(Tf -Tci) = -mhw(Tf -Thi) STEP THREE: Solve (187 kg - mhw)(36 oC - 10oC) = -mhw(36 oC - 50oC) (187 kg - mhw)(26 oC) =-mhw(-14 oC) 4862 oC*kg - 26 oCmhw = 14oCmhw 4862 oC*kg = 14 oCmhw + 26oCmhw 4862 oC*kg / 36 oC = 36oCmhw / 36 oC mhw =135.056 kg

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