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Answer :
To find the specific heat of the metal, we'll use the principle of conservation of energy, which states that the heat lost by the metal will be equal to the heat gained by the water.
Given:
- Mass of metal, [tex]m_{metal} = 0.15 \text{ kg}[/tex]
- Initial temperature of metal, [tex]T_{metal, initial} = 150.0 ^\circ \text{C}[/tex]
- Mass of water, [tex]m_{water} = 0.25 \text{ kg}[/tex]
- Initial temperature of water, [tex]T_{water, initial} = 10.0 ^\circ \text{C}[/tex]
- Final equilibrium temperature, [tex]T_{final} = 22.4 ^\circ \text{C}[/tex]
- Specific heat of water, [tex]c_{water} = 4186 \text{ J/(kg} \cdot \text{°C)}[/tex]
Formula:
The heat lost by the metal can be expressed as:
[tex]Q_{metal} = m_{metal} \times c_{metal} \times (T_{final} - T_{metal, initial})[/tex]
The heat gained by the water can be expressed as:
[tex]Q_{water} = m_{water} \times c_{water} \times (T_{final} - T_{water, initial})[/tex]
Since the heat lost by the metal is equal to the heat gained by the water:
[tex]m_{metal} \times c_{metal} \times (T_{final} - T_{metal, initial}) = m_{water} \times c_{water} \times (T_{final} - T_{water, initial})[/tex]
Substitute the values:
[tex]0.15 \times c_{metal} \times (22.4 - 150.0) = 0.25 \times 4186 \times (22.4 - 10.0)[/tex]
[tex]0.15 \times c_{metal} \times (-127.6) = 0.25 \times 4186 \times 12.4[/tex]
Calculate the right-hand side:
[tex]Q_{water} = 0.25 \times 4186 \times 12.4 = 12972.2 \text{ J}[/tex]
Rearrange and solve for [tex]c_{metal}[/tex]:
[tex]0.15 \times (-127.6) \times c_{metal} = 12972.2[/tex]
[tex]c_{metal} = \frac{12972.2}{0.15 \times (-127.6)}[/tex]
[tex]c_{metal} = \frac{12972.2}{-19.14}[/tex]
[tex]c_{metal} = -678.3 \text{ J/(kg} \cdot \text{°C)}[/tex]
Since specific heat capacity cannot be negative, the negative sign indicates that the equation needs to be corrected for direction, hence:
[tex]c_{metal} = 678.3 \text{ J/(kg} \cdot \text{°C)}[/tex]
Conclusion:
The specific heat of the metal is approximately [tex]678.3 \text{ J/(kg} \cdot \text{°C)}[/tex]. This value tells us how much energy per mass and temperature change the metal requires to heat up or cool down.
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