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:
(1)
Where:
- Mass of ice, in grams.
- Mass of the juice, in grams.
- Specific heat of ice, in joules per gram-degree Celsius.
- Specific heat of water, in joules per gram-degree Celsius.
- Latent heat of fusion, in joules per gram.
- Initial temperature of ice, in degrees Celsius.
- Melting point of water, in degrees Celsius.
- Final temperature of the ice-juice system, in degrees Celsius.
- Initial temperature of the juice, in degrees Celsius.
If we know that
,
,
,
,
,
,
and
, then the mass of the juice is:
![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})}](https://tex.z-dn.net/?f=m_%7Bw%7D%20%3D%20%5Cfrac%7Bm_%7Bi%7D%5Ccdot%20%5Bc_%7Bi%7D%5Ccdot%20%28T_%7B1%7D-T_%7B2%7D%29%20-%20L_%7Bf%7D%20%2B%20c_%7Bw%7D%5Ccdot%20%28T_%7B2%7D-T_%7B3%7D%29%5D%7D%7Bc_%7Bw%7D%20%5Ccdot%20%28T_%7B3%7D-T_%7B4%7D%29%7D)
![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)}](https://tex.z-dn.net/?f=m_%7Bw%7D%20%3D%20%5Cfrac%7B%28100%5C%2Cg%29%5Ccdot%20%20%5Cleft%5B%5Cleft%282.090%5C%2C%5Cfrac%7BJ%7D%7Bg%5Ccdot%20%5E%7B%5Ccirc%7DC%7D%20%5Cright%29%5Ccdot%20%28-10%5C%2C%5E%7B%5Ccirc%7DC%29%20-%20334%5C%2C%5Cfrac%7BJ%7D%7Bg%7D%20%2B%5Cleft%284.18%5C%2C%5Cfrac%7BJ%7D%7Bg%5Ccdot%20%5E%7B%5Ccirc%7DC%7D%20%5Cright%29%5Ccdot%20%28-10%5C%2C%5E%7B%5Ccirc%7DC%29%20%20%5Cright%5D%7D%7B%5Cleft%284.180%5C%2C%5Cfrac%7BJ%7D%7Bg%5Ccdot%20%5E%7B%5Ccirc%7DC%7D%20%5Cright%29%5Ccdot%20%28-10%5C%2C%5E%7B%5Ccirc%7DC%29%7D)

The mass of the juice responsible for melting the ice is 949.043 grams.