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## Homework Statement

(a) In an espresso coffee machine, steam at 100 °C is passed into milk to heat it. Calculate (i) the energy required to heat 150 g of milk from room temperature (20 °C) to 80 °C, (ii) the mass of steam condensed.

(b) A student measures the temperature of the hot coffee as it cools. The results are given below:

A friend suggests that the rate of cooling is exponential. (i) Show quantitatively whether this suggestion is valid. (ii) Estimate the temperature of the coffee after a total of 12 min.

- Specific heat capacity of milk = 4.0 kJ kg
^{-1}K^{-1} - Specific heat capacity of water = 4.2 kJ kg
^{-1}K^{-1} - Specific latent heat of steam = 2.2 MJ kg
^{-1}

^{4}J, (ii) 15.8 g

**2. The attempt at a solution**

(a) (i) ΔQ = mcΔθ = 0.15 kg * 4000 J kg

^{-1}K

^{-1}* (80-20 °C) = 36 000 J -- the energy required to heat 150 g of milk from room temperature of 20 degrees to 80 degrees.

(a) (ii) ΔQ = ml → to find mass m = ΔQ / l = 36 000 J / 2.2 * 10

^{6}J kg

^{-1}= 0.016 kg or 16.4 g. Which is wrong. What did I miss?

(b) (i) Can't say that it's exponential, since the increase is not the same in percentage.

(b) (ii) The temperature should be 29 °C, because we have a difference of two twice and a difference of one once. So, I guess every difference should be two times the number: 2 x 2, 2 x 1, 2 x 0.

In sum: what's missing in (a) (ii) and is my logic in (b) (i-ii) correct?

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