Equilibrium temperature of a mixture

[frazdaz]

Homework Statement

Homework Equations

$$Q = m c \Delta T \\ Q = m L_v \\ Q_{in} = Q_{out}$$

The Attempt at a Solution

$$m_s c_s (100 - 8) + m_s L_v = m_s c_s (100 - T) + m_m c_s (T) \\ 5778.06 + 33900 = 5778.06 - 62.805 T + 418.7 T \\ 33900 = 355.895 T \\ ∴ T = 95.25°C$$
Answer should be 90. This particular working is using 8°C as a base but I've with absolute temperatures and still got the same answer so I'm guessing my logic is flawed?
Thanks

 

[TSny]

Your first equation doesn't appear to be set up correctly. The left side has two terms and the right side has two terms.

Can you describe in words the meaning of each of the four terms in the equation?

 

[frazdaz]

Your first equation doesn't appear to be set up correctly. The left side has two terms and the right side has two terms.

Can you describe in words the meaning of each of the four terms in the equation?

Q_in = extra thermal energy the steam has over the milk + heat to be released when the steam condenses
Q_out = heat lost from the steam + heat gained by milk

 

[SteamKing]

The term

m$_{s}$c$_{s}$(100-8)

implies that the mass of steam cools from 100 C to the temperature of the milk at 8 C after the steam condenses. This is clearly not the case, as the mixture will assume some as yet unknown temperature, which is higher than 8 C once thermal equilibrium is reached.

Rather than use Q$_{in}$ = Q$_{out}$, which is somewhat vague for this type of problem, why not use instead Q$_{lost-from-steam}$ = Q$_{gained-by-milk}$

 

[HalfLostAndSearching]

for your question.

It is important to note that the equilibrium temperature of a mixture is dependent on the specific heat capacities and latent heats of the substances involved, as well as the initial temperatures and masses of each substance. In this case, your calculations seem to be correct, but the final answer of 95.25°C may not be accurate due to rounding errors or other factors. It would be helpful to double check your calculations and consider any other factors that may affect the final result. Additionally, it may be helpful to provide more context or information about the problem in order to accurately assess the validity of your solution.