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Step-by-Step Solution
Step 1: Identify the physical process
An initially hot metal sample (unknown metal) is placed in a brass calorimeter that contains water. Heat flows from the hot metal to both the brass calorimeter and water until thermal equilibrium is reached at a final temperature.
Step 2: Write the heat balance equation
By the principle of calorimetry (neglecting any heat loss to the surroundings), the total heat lost by the hot metal must equal the total heat gained by the calorimeter plus the heat gained by the water. Symbolically:
$ m_{\text{metal}} \, c_{\text{metal}}\, (T_{\text{initial, metal}} - T_{\text{final}})
= m_{\text{cal}} \, c_{\text{brass}} \,(T_{\text{final}} - T_{\text{initial, cal}})
+ m_{\text{water}} \, c_{\text{water}} \,(T_{\text{final}} - T_{\text{initial, water}})\!.$
Step 3: Substitute the given data
Mass of unknown metal, $m_{\text{metal}} = 192 \text{ g}$
Initial temperature of metal, $T_{\text{initial, metal}} = 100^\circ \text{C}$
Mass of brass calorimeter, $m_{\text{cal}} = 128 \text{ g}$
Specific heat of brass, $c_{\text{brass}} = 394 \,\text{J\,kg}^{-1}\,\text{K}^{-1}$
Mass of water, $m_{\text{water}} = 240 \text{ g}$
Specific heat of water, $c_{\text{water}} = 4200 \,\text{J\,kg}^{-1}\,\text{K}^{-1}$
Initial temperature of water and calorimeter, $T_{\text{initial, cal}} = T_{\text{initial, water}} = 8.4^\circ \text{C}$
Final temperature reached by all, $T_{\text{final}} = 21.5^\circ \text{C}$
Unknown specific heat of metal, $c_{\text{metal}} = S \, (\text{J\,kg}^{-1}\,\text{K}^{-1})$
The heat-balance equation (in the form presented in the problem) is:
$192 \times S \times (100 - 21.5)
= 128 \times 394 \times (21.5 - 8.4) + 240 \times 4200 \times (21.5 - 8.4)$.
Step 4: Simplify and solve for $S$
Calculate the temperature differences:
For the metal: $100 - 21.5 = 78.5$
For the brass and water: $21.5 - 8.4 = 13.1$
Write the numerical form:
$192 \times S \times 78.5
= (128 \times 394 \times 13.1) + (240 \times 4200 \times 13.1)$.
Compute the right-hand side terms (or keep them symbolic if desired), then isolate $S$ on the left.
Upon simplification, one obtains:
$S = 916 \,\text{J\,kg}^{-1}\,\text{K}^{-1}$.
Step 5: State the final result
The specific heat capacity of the unknown metal is
916 J kg–1 K–1.
