25.6 {\rm mL} of ethanol (density =0.789 {\rm g}/{\rm mL}) initially at 8.0^\circ {\rm C} is mixed with 39.5 {\rm mL} of water (density = 1.0 {\rm g}/{\rm mL}) initially at 23.1^\circ {\rm C} in an insulated beaker.
Assuming that no heat is lost, what is the final temperature of the mixture?
[mass ethanol x specific heat ethanol x (Tfinal-Tinitial)] + [mass H2O x specific heat H2O x (Tfinal-Tinitial)] = 0
The only unknown is Tf. Solve for that.
so am I treating density as Sp. heat?
Of course not. Use density as in mass = volume x density to convert the volumes in the problem to grams. The specific heat is either given in the problem or it will be in your text/notes. I don't have specific heat tables memorized although I remember that H2O is 4.184 J/g*C or 1 cal/g*C. Use the one you are familiar with.
To find the final temperature of the mixture, we can use the principle of energy conservation. The total heat gained by the water must be equal to the total heat lost by the ethanol. The formula we can use to calculate the heat gained or lost is:
Q = mcΔT
Where:
Q is the heat gained or lost (in joules),
m is the mass (in grams),
c is the specific heat capacity (in J/g°C), and
ΔT is the change in temperature (in °C).
First, let's calculate the mass of ethanol and water:
Mass of ethanol = volume of ethanol x density of ethanol
Mass of ethanol = 25.6 mL x 0.789 g/mL = 20.1984 g
Mass of water = volume of water x density of water
Mass of water = 39.5 mL x 1.0 g/mL = 39.5 g
Next, let's calculate the heat gained or lost by the ethanol and water:
Q_ethanol = m_ethanol x c_ethanol x ΔT_ethanol
Q_water = m_water x c_water x ΔT_water
The specific heat capacity (c) for ethanol is approximately 2.44 J/g°C, and for water, it is 4.18 J/g°C.
Since no heat is lost, Q_ethanol is equal to -Q_water (negative sign indicates loss of heat for ethanol and gain of heat for water). Therefore:
m_ethanol x c_ethanol x ΔT_ethanol = -m_water x c_water x ΔT_water
Simplifying:
(20.1984 g) x (2.44 J/g°C) x (T_f - 8.0°C) = -(39.5 g) x (4.18 J/g°C) x (T_f - 23.1°C)
Now we can solve for T_f, the final temperature of the mixture. Let's solve the equation:
(49.330016 g°C) x (T_f - 8.0°C) = (-164.51 g°C) x (T_f - 23.1°C)
Expand and simplify:
49.330016T_f - 394.640128 = -164.51T_f + 3813.9961
Combine like terms:
213.840016T_f = 4208.636228
Divide both sides by 213.840016:
T_f = 19.67°C
Therefore, the final temperature of the mixture is approximately 19.67°C.