Your hot tub has overheated to a very uncomfortable 48 degrees C. If your hot tub contains 402 L of hot water, what mass of ice at -5 degrees C would you need to add to your hot tub to bring the temperature down to a more comfortable 36 degrees C?

You have to heat the ice to 0

you have to heat to melt the ice at0c
then you have to heat the melted ice to 36C
then you have to "heat" (negative heat) the original water from 48 to 36
the sume of all these heats is zero.

To find the mass of ice needed to bring the temperature of the hot tub down, we can use the principle of heat transfer and the specific heat capacity formula. Here's how to calculate it step by step:

1. Determine the heat transfer required to bring the hot tub temperature down. The formula is:
Q = mcΔT

Q = Heat transfer (in joules)
m = Mass (in kilograms)
c = Specific heat capacity (in joules per kilogram per degree Celsius)
ΔT = Change in temperature (in degrees Celsius)

The specific heat capacity of water is approximately 4,184 J/(kg°C).

2. Calculate the heat transfer required using the initial and final temperatures:
Q = 402 L × 1 kg/L × 4,184 J/(kg°C) × (48°C - 36°C)

Since 1 liter of water is equal to 1 kilogram, we can directly use 402 L as the mass.

Q = 402 kg × 4,184 J/(kg°C) × 12°C

3. Convert the temperature change to Kelvin:
ΔT = 12°C + 273.15 K = 285.15 K

4. Calculate the heat transfer required:
Q = 402 kg × 4,184 J/(kg°C) × 285.15 K

5. The heat lost by the hot water will be gained by the ice used. Assuming the final temperature of the ice-water mixture is 0°C, we can write:
Q = mcΔT

Where "m" is the mass of ice needed (in kilograms), "c" is the specific heat capacity of ice (approximately 2,093 J/(kg°C)), and "ΔT" is the temperature change from -5°C to 0°C.

6. Calculate the mass of ice needed:
Q = m × 2,093 J/(kg°C) × (0°C - (-5°C))

Q = m × 2,093 J/(kg°C) × 5°C

Equating the two expressions for Q, we can solve for "m".

7. Set up the equation:
402 kg × 4,184 J/(kg°C) × 285.15 K = m × 2,093 J/(kg°C) × 5°C

8. Solve for "m":
m = (402 kg × 4,184 J/(kg°C) × 285.15 K) / (2,093 J/(kg°C) × 5°C)

Calculate the right side of the equation to find the mass of ice needed.

After following these steps and calculating the mass of ice needed, you will have the answer to your question.