A well-insulated

PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Thermodynamics · Thermodynamics

Problem

A well-insulated, rigid calorimeter contains two immiscible liquids that do not mix chemically but remain in good thermal contact through their interface. Liquid X has mass

m_{\text{X}}

and specific heat

c_{\text{X}}

and is initially at temperature

T_{\text{X0}}

. Liquid Y has mass

m_{\text{Y}}

and specific heat

c_{\text{Y}}

and is initially at temperature

T_{\text{Y0}}

. Immediately after the liquids are poured into the calorimeter they form two horizontal layers with an interface area that is large compared with the thickness of either layer. Neglect any energy transfer to the calorimeter walls. (a) Derive an algebraic expression for the common equilibrium temperature

T_{\text{eq}}

of the two-liquid system in terms of

m_{\text{X}},\,m_{\text{Y}},\,c_{\text{X}},\,c_{\text{Y}},\,T_{\text{X0}},

and

T_{\text{Y0}}

. Clearly state the physical principle used. (b) Explain why the net energy transfer across the interface must be zero once the temperature

T_{\text{eq}}

you derived is achieved. (c) Suppose a small electric heater supplies energy

Q_{\text{in}}

uniformly to liquid X after equilibrium has been established. Determine the new equilibrium temperature

T^{\,*}_{\text{eq}}

that will eventually be reached, expressing your answer in terms of

Q_{\text{in}}

and the other given variables. Neglect energy losses to the surroundings. (d) Discuss qualitatively how the magnitude of the interface area influences the time required to reach the original equilibrium temperature

T_{\text{eq}}

derived in part (a). Do not perform a numerical calculation; base your reasoning on molecular-level interactions and the macroscopic definition of thermal equilibrium.

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