Question

Sarah is making bread. She measures 0.6 kg of hot water into a bowl. The water needs to be at a temperature of
exactly 42°C in order for the yeast to function properly, but right now it is at 40°C. What mass of additional
water at 49°C should she add to make it the correct temperature? Assume no heat is transferred to the bowl or
to the outside air. And please show your work

Answer 1

Answer:

$\Delta m = 0.171\,kg$

Explanation:

The process of water adding is described by the First Law of Thermodynamics:

$m_{w,o} \cdot h_{w,o} + \Delta m \cdot h_{w} = (m_{w,o} + \Delta m)\cdot h_{w,f}$

The amount of additional mass is:

$m_{w,o}\cdot h_{w,o} - m_{w,o}\cdot h_{w,f} = \Delta m\cdot (h_{w,f}-h_{w})$

$\Delta m = \frac{m_{w,o}\cdot(h_{w,o}-h_{w,f})}{h_{w,f}-h_{w}}$

Given that water is incompressible, the equation can be further simplified:

$\Delta m = m_{w,o}\cdot \frac{c_{p,w}\cdot (T_{w,o}-T_{w,f})}{c_{p,w}\cdot (T_{w,f}-T_{w})}$

$\Delta m = m_{w,o}\cdot \left(\frac{T_{w,o}-T_{w,f}}{T_{w,f}-T_{w}} \right)$

$\Delta m = (0.6\,kg)\cdot \left(\frac{40 ^{\circ}C - 42^{\circ}C}{42^{\circ}C-49^{\circ}C} \right)$

$\Delta m = 0.171\,kg$

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