A hot tub is filled with 410 gallons of water. If the hot tub heater provides 5900 kJ/min, how long (in hours) will it take to heat the water in the hot tub from 67 degrees Fahrenheit to 197 degrees Fahrenheit?
Here are the steps you need.
1. Convert 410 gallons of water to liters.
2. Using density of water as 1.0 g/mL, convert to grams H2O
3. Convert 67 degrees F to degrees celsius
4. Convert 197 degrees F to degrees Celsius
5. q = mass H2O in grams x specific heat H2O x (Tfinal-Tintial) knowing specific heat H2O is 4.184 J/g*C
6. Convert q in J to kJ
7. Then use kJ/min x # min = kJ in 6
Post your work if you get stuck.
I'm not an engineer so I don't know specific heat in J/gallon*degrees F and couldn't find it on the web therefore I converted everything to grams and degrees Celsius
Post your work if you get stuck. .
To calculate the time it will take to heat the water in the hot tub, we need to use the formula:
𝑄 = 𝑚𝐶∆𝑇
Where:
𝑄 = heat energy (in joules)
𝑚 = mass of water (in kg)
𝐶 = specific heat capacity of water (in J/kg°C)
∆𝑇 = change in temperature (in °C)
Let's follow these steps to find the time it will take to heat the water:
Step 1: Convert gallons to liters and then to kilograms.
1 gallon ≈ 3.78541 liters
410 gallons ≈ 410 × 3.78541 liters
410 gallons ≈ 1549.5081 liters
Density of water = 1 kg/L
Mass of water = 1549.5081 kg
Step 2: Calculate the change in temperature.
∆𝑇 = 197°C - 67°C
∆𝑇 = 130°C
Step 3: Determine the specific heat capacity of water.
The specific heat capacity of water is approximately 4186 J/kg°C.
Step 4: Calculate the heat energy required.
𝑄 = 𝑚𝐶∆𝑇
𝑄 = 1549.5081 kg × 4186 J/kg°C × 130°C
𝑄 = 116,202,146.2306 J
Step 5: Convert the heat energy to kilojoules.
1 kJ = 1000 J
𝑄 = 116,202,146.2306 J / 1000
𝑄 = 116,202.1462306 kJ
Step 6: Calculate the time required using the heating power.
Heating power = 5900 kJ/min
Time = 𝑄 / heating power
Time = 116,202.1462306 kJ / 5900 kJ/min
Step 7: Convert minutes to hours.
1 hour = 60 minutes
Time (in hours) = (116,202.1462306 kJ / 5900 kJ/min) / 60 min/hr
Now, let's calculate the time it will take:
Time (in hours) = 116,202.1462306 / (5900 × 60)
Time (in hours) ≈ 3.15 hours
Therefore, it will take approximately 3.15 hours to heat the water in the hot tub from 67°F to 197°F.
To find the time it takes to heat the water in the hot tub, we need to calculate the amount of heat energy required and then divide it by the heater output rate.
First, let's calculate the heat energy required to heat the water. The formula to calculate heat energy is Q = m * c * ΔT, where Q is the heat energy, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.
1. Calculate the mass of water: Since the hot tub is filled with 410 gallons of water, we need to convert gallons to liters (1 gallon ≈ 3.78541 liters):
410 gallons * 3.78541 liters/gallon = 1550.9471 liters
2. Convert the mass of water to kilograms (1 liter of water ≈ 1 kilogram):
1550.9471 liters = 1550.9471 kilograms
3. Calculate the change in temperature:
ΔT = final temperature - initial temperature
ΔT = (197°F - 67°F) = 130°F
4. Convert the change in temperature to Celsius (1°F ≈ 0.556°C):
ΔT = 130°F * 0.556°C/°F = 72.28°C
5. Calculate the heat energy required using the specific heat capacity of water:
c = 4.18 J/g°C (specific heat capacity of water)
Q = m * c * ΔT
Q = 1550.9471 kg * 4.18 J/g°C * 72.28°C
Q = 447,624.8467 kJ
Now that we have calculated the heat energy required, we can find the time it takes using the heater output rate.
6. Divide the heat energy required by the heater output rate:
Time = Heat energy required / Heater output rate
Time = 447,624.8467 kJ ÷ 5900 kJ/min
Finally, convert the time from minutes to hours:
Time = (447,624.8467 kJ ÷ 5900 kJ/min) / 60 min/hour
Calculate the final value:
Time = 12.986 hours (rounded to three decimal places)
Therefore, it will take approximately 12.986 hours to heat the water in the hot tub from 67 degrees Fahrenheit to 197 degrees Fahrenheit.