If a hot iron ball is dropped into 200.0 gram of cooler waterthe water temperature increases by 2.0 celsius and the temperature of the ball decreases by 18.6 celsius. What is the mass?

To solve this problem, we can use the principle of conservation of energy, specifically the principle of heat transfer. The amount of heat gained by the water is equal to the amount of heat lost by the iron ball.

The heat gained by the water, expressed in calories, is given by the formula:

Q_water = mass_water * specific_heat_water * change_in_temperature_water

Where:
- mass_water is the mass of the water in grams (given as 200.0 g)
- specific_heat_water is the specific heat capacity of water, which is 1.00 cal/g·°C
- change_in_temperature_water is the increase in temperature of the water, which is 2.0 °C

The heat lost by the iron ball, expressed in calories, is given by the formula:

Q_ball = mass_ball * specific_heat_ball * change_in_temperature_ball

Where:
- mass_ball is the mass of the iron ball, which we are trying to find
- specific_heat_ball is the specific heat capacity of iron, which is 0.11 cal/g·°C
- change_in_temperature_ball is the decrease in temperature of the iron ball, which is -18.6 °C

According to the principle of heat transfer, Q_water = -Q_ball (since the heat gained by the water is equal to the heat lost by the iron ball). Therefore, we can set up the following equation:

mass_water * specific_heat_water * change_in_temperature_water = mass_ball * specific_heat_ball * change_in_temperature_ball

Substituting the known values:

200.0 g * 1.00 cal/g·°C * 2.0 °C = mass_ball * 0.11 cal/g·°C * (-18.6 °C)

Simplifying the equation:

400.0 cal = -mass_ball * 2.046 cal

Now, isolate mass_ball by dividing both sides of the equation by -2.046 cal:

mass_ball = 400.0 cal / (-2.046 cal/g)

mass_ball = -195.78 g

Since mass cannot be negative, the mass of the iron ball is 195.78 grams.