The water in your bathtub is getting cold. You currently have 36 liters at 31°C. What is the new temperature after you add 10 liters of 70°C hot water?
Degrees Celsius are entered as degC.
To find the new temperature of the water in the bathtub after adding hot water, we can use the principle of the conservation of energy, specifically the principle of heat transfer.
The equation for heat transfer is Q = mcΔT, where:
- Q is the heat transferred (in joules or calories),
- m is the mass of the water (in kilograms or grams),
- c is the specific heat capacity of water (4.18 J/g°C or 1 cal/g°C), and
- ΔT is the change in temperature (in degrees Celsius).
First, let's find the initial heat energy (Q1) of the water in the bathtub using the equation Q = mcΔT:
Q1 = (mass1) × (specific heat capacity of water) × (ΔT1)
Q1 = (36 kg) × (4.18 J/g°C) × (31°C - initial temperature of water)
Next, let's find the heat energy (Q2) of the hot water being added using the equation Q = mcΔT:
Q2 = (mass2) × (specific heat capacity of water) × (ΔT2)
Q2 = (10 kg) × (4.18 J/g°C) × (70°C - initial temperature of hot water)
Since the heat energy gained by the water in the bathtub should be equal to the heat energy lost by the hot water, we can set up an equation:
Q1 + Q2 = 0
Substituting the values, we have:
(36 kg) × (4.18 J/g°C) × (31°C - initial temperature of water) + (10 kg) × (4.18 J/g°C) × (70°C - initial temperature of hot water) = 0
Now, rearrange the equation and solve for the initial temperature of water:
(36 kg) × (4.18 J/g°C) × (31°C - initial temperature of water) = - (10 kg) × (4.18 J/g°C) × (70°C - initial temperature of hot water)
Solving for the initial temperature of water, we get:
(36 kg) × (4.18 J/g°C) × (31°C - initial temperature of water) = (10 kg) × (4.18 J/g°C) × (70°C - initial temperature of hot water)
Simplifying the equation gives:
(36 kg) × (31°C - initial temperature of water) = (10 kg) × (70°C - initial temperature of hot water)
Now, we can solve for the initial temperature of water.