Consider a perfectly insulated container that initially holds three separate solid…

CHEMISTRY · Multi-part · A Level · Chemistry · Energetics · Energetics

Problem

Consider a perfectly insulated container that initially holds three separate solid objects—A, B, and C—each made of a different substance. Their masses

(m)

, molar heat capacities at constant pressure

(C_{p,m})

, and initial temperatures

(T_0)

are listed below. Object A:

m_A = 0.200\,\text{kg}

C_{p,m,A}=24\,\text{J}\,\text{mol}^{-1}\,^{\circ}\text{C}^{-1}

T_{0,A}=350\,^{\circ}\text{C}

M_A = 56.0\,\text{g}\,\text{mol}^{-1}

Object B:

m_B = 0.500\,\text{kg}

C_{p,m,B}=29\,\text{J}\,\text{mol}^{-1}\,^{\circ}\text{C}^{-1}

T_{0,B}=50\,^{\circ}\text{C}

M_B = 64.0\,\text{g}\,\text{mol}^{-1}

Object C:

m_C = 0.300\,\text{kg}

C_{p,m,C}=27\,\text{J}\,\text{mol}^{-1}\,^{\circ}\text{C}^{-1}

T_{0,C}=20\,^{\circ}\text{C}

M_C = 88.0\,\text{g}\,\text{mol}^{-1}

Assume heat capacities remain constant and that no phase changes occur. (1) Derive a general symbolic expression for the final equilibrium temperature

T_f

n

objects in an insulated system, expressing

T_f

solely in terms of the masses

m_i

, molar masses

M_i

, molar heat capacities

C_{p,m,i}

, and initial temperatures

T_{0,i}

of the objects. (2) Use your expression to calculate the numerical value of

T_f

for objects A, B, and C listed above. Show all steps and report

T_f

to the nearest

0.1^{\circ}\text{C}

. (3) Suppose a fourth object D with an extremely large value of

m\,C_p

(effectively an infinite thermal capacity) is introduced at

25\,^{\circ}\text{C}

and simultaneously brought into contact with A, B, and C inside the same insulated container. Using your expression from part (a), determine the new equilibrium temperature and explain the physical significance of your result.

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