Question

An ice-making machine inside a refrigerator operates in a Carnot cycle. It takes heat from liquid water at 0.0 degrees Celsius and rejects heat to a room at a temperature of 19.2 degrees Celsius. Suppose that liquid water with a mass of 74.6kg at 0.0 degrees Celsius is converted to ice at the same temperature. Take the heat of fusion for water to be L_f = 3.34*10^5 J/kg.
How much energy E must be supplied to the device?

Express your answer in joules

Answer 1

Answer:

1751399.8902 J

Explanation:

m = Mass of water = 74.6 kg

$\Delta H_f$ = Latent heat of fusion = $3.34\times 10^5\ J/kg$

$T_h$ = 19.2°C

$T_c$ = 0°C

We have the equation for cold body

$Q_c=m\Delta H_f$

We have the equation

$\dfrac{Q_c}{Q_h}=\dfrac{T_c}{T_h}\\\Rightarrow Q_h=\dfrac{Q_cT_h}{T_c}\\\Rightarrow Q_h=\dfrac{m\Delta H_fT_h}{T_c}\\\Rightarrow Q_h=\dfrac{74.6\times 3.34\times 10^5\times (19.2+273.15)}{0+273.15}\\\Rightarrow Q_h=26667799.8902\ J$

Energy rejected to the room is 26667799.8902 J

Energy supplied to the device is given by

$E=Q_h-m\Delta H_f\\\Rightarrow E=26667799.8902-74.6\times 3.34\times 10^5\\\Rightarrow E=1751399.8902\ J$

The energy supplied to the device is 1751399.8902 J