Problem 3

In an insulated vessel,

m_h=0.40\,\mathrm{kg}

of water at

60^\circ\mathrm C

is mixed with

m_c=0.20\,\mathrm{kg}

of water at

30^\circ\mathrm C

. Water has specific heat

c=4200\,\mathrm{J\,kg^{-1}K^{-1}}

; neglect the vessel heat capacity and heat loss. After mixing, the water reaches one uniform final temperature, with no evaporation or phase change. Heat is exchanged only between the two portions of water; the insulated vessel takes up no energy. Define heat as positive when the portion considered gains energy and negative when it loses energy. (a) Find the equilibrium temperature

T_f

. (b) Calculate the signed heat transferred by the hot water. (c) Find the heat received by the cold water and check energy conservation. (d) State the ordering of

T_c,T_f,T_h

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

T_f=\frac{m_hT_h+m_cT_c}{m_h+m_c}=50.0^\circ\mathrm C

problems.proof.analysis

Solve the heat-balance equation for the final temperature; it is a mass-weighted average of the initial temperatures. The result must lie between them.

problems.proof.pitfall. The hot water’s signed heat is negative because it loses energy; use one consistent sign convention in the heat balance.