Answer:
The mass of the juice responsible for melting the ice is 949.043 grams.
Explanation:
By the First Law of Thermodynamics, we understand that juice releases heat to the ice, which turns into water under the assumption that interactions between the ice-juice system and surroundings are negligible and energy processes are done in steady-state. Since juice is done with water, its specific heat will be taken as of the water. The process is described by the following formula:
[tex]m_{i} \cdot [c_{i}\cdot (T_{1}-T_{2}) - L_{f} + c_{w}\cdot (T_{2}-T_{3})] + m_{w} \cdot c_{w}\cdot (T_{4}-T_{3}) = 0[/tex] (1)
Where:
[tex]m_{i}[/tex] - Mass of ice, in grams.
[tex]m_{w}[/tex] - Mass of the juice, in grams.
[tex]c_{i}[/tex] - Specific heat of ice, in joules per gram-degree Celsius.
[tex]c_{w}[/tex] - Specific heat of water, in joules per gram-degree Celsius.
[tex]L_{f}[/tex] - Latent heat of fusion, in joules per gram.
[tex]T_{1}[/tex] - Initial temperature of ice, in degrees Celsius.
[tex]T_{2}[/tex] - Melting point of water, in degrees Celsius.
[tex]T_{3}[/tex] - Final temperature of the ice-juice system, in degrees Celsius.
[tex]T_{4}[/tex] - Initial temperature of the juice, in degrees Celsius.
If we know that [tex]m_{i} = 100\,g[/tex], [tex]c_{i} = 2.090\,\frac{J}{g\cdot ^{\circ}C}[/tex], [tex]c_{w} = 4.18\,\frac{J}{g\cdot ^{\circ}C}[/tex], [tex]L_{f} = 334\,\frac{J}{g}[/tex], [tex]T_{1} = -10\,^{\circ}C[/tex], [tex]T_{2} = 0\,^{\circ}C[/tex], [tex]T_{3} = 10\,^{\circ}C[/tex] and [tex]T_{4} = 20\,^{\circ}C[/tex], then the mass of the juice is:
[tex]m_{w} = \frac{m_{i}\cdot [c_{i}\cdot (T_{1}-T_{2}) - L_{f} + c_{w}\cdot (T_{2}-T_{3})]}{c_{w} \cdot (T_{3}-T_{4})}[/tex]
[tex]m_{w} = \frac{(100\,g)\cdot \left[\left(2.090\,\frac{J}{g\cdot ^{\circ}C} \right)\cdot (-10\,^{\circ}C) - 334\,\frac{J}{g} +\left(4.18\,\frac{J}{g\cdot ^{\circ}C} \right)\cdot (-10\,^{\circ}C) \right]}{\left(4.180\,\frac{J}{g\cdot ^{\circ}C} \right)\cdot (-10\,^{\circ}C)}[/tex]
[tex]m_{w} = 949.043\,g[/tex]
The mass of the juice responsible for melting the ice is 949.043 grams.
