A 63.8 kg block of silver is at initially at 107 C. 1.0 kg of ice at its freezing point is placed onto the block to cool it. As the ice melts, the water is free to flow off (so that the water never increases in temperature). After all the ice melts, what is the final temperature of the silver? Round answers to the nearest whole number, and in degrees Celsius.

Please see other post, and check the heat capacity/other constants.

To find the final temperature of the silver block, we need to consider the heat exchanged between the silver block and the ice.

First, let's calculate the heat gained by the ice from cooling down to its freezing point:

q1 = m * c * ΔT

Where:
m = mass of the ice = 1.0 kg
c = specific heat capacity of ice = 2090 J/kg°C (approximately)
ΔT = change in temperature = 0°C (reaching the freezing point)

q1 = 1.0 kg * 2090 J/kg°C * 0°C = 0 J (no change in temperature)

The heat gained by the ice is 0 J since it doesn't increase in temperature at its freezing point.

Now, let's calculate the heat lost by the silver block as it cools down from 107°C to the final temperature:

q2 = m * c * ΔT

Where:
m = mass of the silver block = 63.8 kg
c = specific heat capacity of silver = 235 J/kg°C (approximately)
ΔT = change in temperature = 107°C - 𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 (since the water never increases in temperature)

The specific heat capacity of silver is lower than that of ice, so the effect on temperature will be greater for the silver block than for the ice. Therefore, the final temperature of the silver will be less than the freezing point of water.

Now we can solve for the final temperature of the silver using the heat equation:

- q1 = q2

0 J = (63.8 kg) * (235 J/kg°C) * (107°C - 𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)

Simplifying the equation:

0 J = 16057.3 J/°C * (107°C - 𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)
0 J = 16057.3 J/°C * (107°C - 𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)

Dividing both sides by 16057.3 J/°C:

0°C = 107°C - 𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒

Now, subtracting 107°C from both sides:

-107°C = -𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒

Finally, multiplying both sides by -1:

107°C = 𝐹𝑖𝑛𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒

So, the final temperature of the silver block is 107°C.

Note: The silver doesn't reach a temperature lower than 0°C because the ice and resulting water absorb heat until they reach their melting point, then they flow off. The remaining heat from the silver is not enough to lower its temperature below the freezing/melting point of water.