When you take a bath, how many kilograms of hot water (50°C) must you mix with cold water (13°C) so that the temperature of the bath is 36°C? The total mass of water (hot plus cold) is 215kg. Ignore any heat flow between the water and its external surroundings.
Ah, the classic hot bath dilemma! Let's dive into it, shall we?
To find the amount of hot water needed, we can set up an equation based on the principle of heat conservation. The heat gained by the cold water should be equal to the heat lost by the hot water when they reach the final temperature:
(mass of hot water) x (temperature change for hot water) = (mass of cold water) x (temperature change for cold water)
Now, let's plug in the given values:
(mass of hot water) x (36°C - 50°C) = (mass of cold water) x (36°C - 13°C)
Simplifying the equation:
(mass of hot water) x (-14°C) = (mass of cold water) x (23°C)
Since the total mass of water is 215kg, we can rewrite the equation:
(mass of hot water) x (-14°C) = (215kg - mass of hot water) x (23°C)
Alright, now let's sprinkle in some algebra magic:
-14°C x (mass of hot water) = 23°C x (215kg - mass of hot water)
Expanding the equation:
-14°C x mass of hot water = 23°C x 215kg - 23°C x mass of hot water
Now, let's solve for the mass of hot water:
-14°C x mass of hot water + 23°C x mass of hot water = 23°C x 215kg
Combining like terms:
9°C x mass of hot water = 23°C x 215kg
Dividing by 9°C:
mass of hot water = (23°C x 215kg) / 9°C
Doing some final calculations:
mass of hot water ≈ 557.78kg
So, you'll need approximately 557.78 kilograms of hot water to mix with cold water to achieve a cozy 36°C bath! Just be careful not to turn yourself into a human tea bag while measuring it. Safety first!
To find the amount of hot water needed to mix with cold water in order to reach a desired temperature, we need to use the principle of the conservation of energy.
The heat gained by the cold water plus the heat gained by the hot water equals the total heat capacity of the system. Assuming no heat flow with the surroundings, we can write:
Heat gained by cold water + Heat gained by hot water = Total heat capacity of the system
The heat gained or lost by a substance can be calculated using the following formula:
Q = m * c * ΔT
Where:
Q = Heat gained or lost (in Joules)
m = Mass of the substance (in kg)
c = Specific heat capacity of the substance (in J/kg°C)
ΔT = Change in temperature (in °C)
In this case, we need to find the mass of hot water (m_hot) that needs to be mixed with the cold water. Since the specific heat capacity of water is the same for both hot and cold water, we can write the equation as:
(m_hot * c * ΔT_hot) + (m_cold * c * ΔT_cold) = (m_hot + m_cold) * c * ΔT_sys
Where:
m_hot = Mass of hot water (unknown)
m_cold = Mass of cold water (given as the total mass of water - m_hot)
ΔT_hot = Temperature change for hot water (36°C - 50°C = -14°C, as it loses temperature)
ΔT_cold = Temperature change for cold water (36°C - 13°C = 23°C, as it gains temperature)
ΔT_sys = Temperature change for the system (36°C - initial temperature = 36°C - 50°C = -14°C, as it loses temperature)
c = Specific heat capacity of water (assumed to be constant)
Substituting the known values:
(m_hot * c * -14) + ((215 - m_hot) * c * 23) = (215) * c * -14
Simplifying the equation, we can solve for m_hot:
-14m_hot + 23(215 - m_hot) = -14(215)
Solving this equation will give the mass of hot water (m_hot) needed to mix with the cold water.