Equal volumes of three liquids of densities ρ1,ρ2 and ρ3 , specific heat capacities c1,c2 and c3 and temperatures, t1,t2 and t3 respectively are mixed together , What is the temperature of the mixture ? Assume no change in volume on mixing
heat= ρ1*V*c1*(T-t1) + ρ2*c2*(T-t2) + ρ3*c3*V*(T-t3)
but heat=0 as the sum of the heats iz zero (some gain, some lose).
solve for T
To find the temperature of the mixture, we can use the principle of conservation of energy. The heat gained by the cooler liquids must be equal to the heat lost by the hotter liquids.
The heat gained or lost by a liquid is given by the equation:
Q = mcΔt,
where Q is the heat gained or lost, m is the mass of the liquid, c is the specific heat capacity of the liquid, and Δt is the change in temperature.
Let's assume that the volume of each liquid is V, and the mass of each liquid is m (since we are assuming equal volumes and no change in volume on mixing).
The heat gained or lost by each liquid can be expressed as follows:
Q1 = mc1Δt1
Q2 = mc2Δt2
Q3 = mc3Δt3
Since the three liquids are mixed together, the total heat gained by the cooler liquids must be equal to the total heat lost by the hotter liquids:
Q1 + Q2 + Q3 = 0
mc1Δt1 + mc2Δt2 + mc3Δt3 = 0
Since the volumes are equal and there is no change in volume on mixing, the masses of the liquids will also be equal. Let's assume the common mass to be m.
mc1Δt1 + mc2Δt2 + mc3Δt3 = 0
Now, divide both sides of the equation by m:
c1Δt1 + c2Δt2 + c3Δt3 = 0
Now, rearrange the equation to solve for the temperature of the mixture:
c1Δt1 + c2Δt2 + c3Δt3 = 0
Δt1/c1 + Δt2/c2 + Δt3/c3 = 0
Δt1/c1 = - (Δt2/c2 + Δt3/c3)
Now, divide both sides of the equation by (Δt2/c2 + Δt3/c3):
Δt1/c1 / (Δt2/c2 + Δt3/c3) = -1
Finally, solve for Δt1 to find the temperature of the mixture:
Δt1 = - (Δt2/c2 + Δt3/c3) * (c1/c1)
Since the Δt values are the differences between the original temperatures and the final temperature of the mixture, we can rearrange the equation to find the final temperature of the mixture (Tmixture):
Tmixture = Tf - Δt1
where Tf is the final temperature of the mixture (unknown) and Δt1 is the change in temperature for the first liquid:
Tmixture = Tf - (- (Δt2/c2 + Δt3/c3) * (c1/c1))
Simplifying further:
Tmixture = Tf + (Δt2/c2 + Δt3/c3) * (c1/c1)
Tmixture = Tf + (Δt2/c2 + Δt3/c3)
Therefore, the final temperature of the mixture (Tmixture) can be found by summing the changes in temperature for each liquid divided by their respective specific heat capacities.