A hot solid of mass 100 g at 100°c is quickly transferred into 100 g of water in a container of mass 200 g at 20°C. Calculate the resulting temperature of the mixture. Specific heat capacity of the solid and the container is 400 J/kgK.
heat gained/lost = mass * specific heat * temperature change
the heat lost by the "hot solid" is gained by the water and container
To solve this question, we can apply the principle of conservation of energy.
First, let's calculate the heat gained by the container and water:
Q1 = mass_container * specific_heat_capacity * ΔT_container
Q2 = mass_water * specific_heat_capacity * ΔT_water
Where:
- Q1 is the heat gained by the container and water
- mass_container is the mass of the container (200 g)
- specific_heat_capacity is the specific heat capacity of the container (400 J/kgK)
- ΔT_container is the change in temperature of the container (final temperature - initial temperature)
- mass_water is the mass of the water (100 g)
- ΔT_water is the change in temperature of the water (final temperature - initial temperature)
Next, let's consider the heat lost by the hot solid:
Q3 = mass_solid * specific_heat_capacity * ΔT_solid
Where:
- Q3 is the heat lost by the hot solid
- mass_solid is the mass of the hot solid (100 g)
- specific_heat_capacity is the specific heat capacity of the solid and container (400 J/kgK)
- ΔT_solid is the change in temperature of the solid (final temperature - initial temperature)
According to the conservation of energy, the heat gained is equal to the heat lost:
Q1 + Q2 = Q3
Substituting the equations, we get:
mass_container * specific_heat_capacity * ΔT_container + mass_water * specific_heat_capacity * ΔT_water = mass_solid * specific_heat_capacity * ΔT_solid
Given:
- mass_container = 200 g
- specific_heat_capacity = 400 J/kgK
- mass_water = 100 g
- initial temperature of the container = 20°C
- initial temperature of the hot solid = 100°C
Let's calculate the change in temperature of the container (ΔT_container) and the change in temperature of the water (ΔT_water):
ΔT_container = Tf - Ti
= Tf - 20°C
ΔT_water = Tf - Ti
= Tf - 20°C
Now, let's substitute the values in the equation:
200 g * 400 J/kgK * (Tf - 20°C) + 100 g * 400 J/kgK * (Tf - 20°C) = 100 g * 400 J/kgK * (Tf - 100°C)
Simplifying and solving for Tf:
200 Tf - 4000 + 100 Tf - 2000 = 100 Tf - 10000
300 Tf - 6000 = 100 Tf - 10000
200 Tf = 4000
Tf = 4000 / 200
Tf = 20°C
Therefore, the resulting temperature of the mixture is 20°C.
To calculate the resulting temperature of the mixture, we can use the principle of conservation of energy. The energy lost by the hot solid will be gained by the water and the container. The formula we will use is:
mcΔT = mcΔT
Where:
m = mass
c = specific heat capacity
ΔT = change in temperature
Let's break down the given information:
- Mass of the hot solid = 100 g
- Specific heat capacity of the solid = 400 J/kgK
- Temperature of the hot solid = 100°C
- Mass of the water = 100 g
- Mass of the container = 200 g
- Temperature of the water and container = 20°C
First, let's calculate the energy lost by the hot solid:
Energy lost by the solid = mcΔT
The initial temperature of the solid is 100°C, and we want to find the final temperature. So, the change in temperature (ΔT) for the solid will be:
ΔT = Final temperature - Initial temperature
Next, let's calculate the energy gained by the water and the container:
Energy gained by the water and container = (mW + mC)cΔT
Where mW is the mass of water and mC is the mass of the container.
Since the energy lost by the solid is equal to the energy gained by the water and container, we can set up the equation:
(mcΔT)lost = (mW + mC)cΔT
Now let's substitute the values we know into the equation:
(100 g)(400 J/kgK)(ΔT)lost = ((100 g + 200 g)(400 J/kgK)(ΔT)mixture
Now, simplify the equation further:
(100 g)(400 J/kgK)(ΔT)lost = (300 g)(400 J/kgK)(ΔT)mixture
The mass and specific heat capacity cancel out from both sides of the equation, leaving us with:
(ΔT)lost = (ΔT)mixture
This means that both sides have the same value of change in temperature. So, we can simply equate the two:
(ΔT)mixture = (ΔT)lost
Now, solve for the final temperature of the mixture:
100°C - Temperature of the mixture = (ΔT)lost
To find the resulting temperature, subtract the value of (ΔT)lost from 100°C.
Finally, plug in the value of (ΔT)lost into the equation and calculate the resulting temperature of the mixture.