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You wish to cool a 1.83 kg block of tin initially at 88.0°C to a temperature of 57.0°C by placing it in a container of kerosene initially at 24.0°C. Determine the volume (in L) of the liquid needed in order to accomplish this task without boiling. The density and specific heat of kerosene are respectively 820 kg/m3 and 2,010 J/(kg · °C), and the specific heat of tin is 218 J/(kg · °C).

Sagot :

xero099

Answer:

0.273 liters are needed to accomplish this task without boiling.

Explanation:

The minimum boiling point of kerosene is [tex]150\,^{\circ}C[/tex]. According to this question, we need to determine the minimum volume of liquid such that heat received is entirely sensible, that is, with no phase change.

If we consider a steady state process and that energy interactions with surrounding are negligible, then we get the following formula by the Principle of Energy Conservation:

[tex]\rho_{k}\cdot V_{k}\cdot c_{k}\cdot (T-T_{k,o}) = m_{t}\cdot c_{t}\cdot (T_{t,o}-T)[/tex] (1)

Where:

[tex]\rho_{k}[/tex] - Density of kerosene, measured in kilograms per cubic meter.

[tex]V_{k}[/tex] - Volume of kerosene, measured in cubic meters.

[tex]c_{k}[/tex], [tex]c_{t}[/tex] - Specific heats of the kerosene and tin, measured in joule per kilogram-Celsius.

[tex]T_{k,o}[/tex], [tex]T_{t,o}[/tex] - Initial temperatures of kerosene and tin, measured in degrees Celsius.

[tex]T[/tex] - Final temperatures of the kerosene-tin system, measured in degrees Celsius.

Please notice that the block of tin is cooled at the expense of the temperature of the kerosene until thermal equilibrium is reached.

From (1), we clear the volume of kerosene:

[tex]V_{k} = \frac{m_{t}\cdot c_{t}\cdot (T_{t,o}-T)}{\rho_{k}\cdot c_{k}\cdot (T-T_{k,o})}[/tex]

If we know that [tex]m_{t} = 1.83\,kg[/tex], [tex]c_{t} = 218\,\frac{J}{kg\cdot ^{\circ}C}[/tex], [tex]T_{t,o} = 88\,^{\circ}C[/tex], [tex]T_{k,o} = 24.0\,^{\circ}C[/tex], [tex]T = 57\,^{\circ}C[/tex], [tex]c_{k} = 2010\,\frac{J}{kg\cdot ^{\circ}C}[/tex] and [tex]\rho_{k} = 820\,\frac{kg}{m^{3}}[/tex], then the volume of the liquid needed to accomplish this task without boiling is:

[tex]V_{k} = \frac{(1.83\,kg)\cdot \left(218\,\frac{J}{kg\cdot ^{\circ}C} \right)\cdot (88\,^{\circ}C-57\,^{\circ}C)}{\left(820\,\frac{kg}{m^{3}} \right)\cdot \left(2010\,\frac{J}{kg\cdot ^{\circ}C} \right)\cdot (57\,^{\circ}C-24\,^{\circ}C)}[/tex]

[tex]V_{k} = 2.273\times 10^{-4}\,m^{3}[/tex]

[tex]V_{k} = 0.273\,L[/tex]

0.273 liters are needed to accomplish this task without boiling.

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