## Ice Diet

### Task number: 1279

Water does not provide the human body with any usable energy. However, when heating some ice to body temperature, the human body burns off energy (energy needs to be supplied). How many ice cubes with a mass of 50 g and a temperature of −20 °C do you have to eat to lose five kilograms of fat? Burning one gram of fat during metabolic processes in the body releases the heat of 38 kJ.

#### Hint

The energy released by burning the body fat is used for

- heating the ice to the melting point,
- melting the ice,
- heating the melted water to the body temperature.

#### Numerical values

*m*_{1}= 50 g = 0.050 kgthe mass of an ice cube *t*_{I}= −20 °Cthe temperature of ice *m*_{f}= 5 kgthe mass of burnt fat *H*= (38 kJ)/(1 g) = 38×10^{6}Jkg^{−1}the energy released by burning fat *N*= ?the number of ice cubes Other necessary values:

*t*_{B}= 36.6 °Cthe body temperature *t*_{m}= 0 °Cthe melting point of ice *c*_{I}= 2.1 kJkg^{−1}K^{−1}= 2100 Jkg^{−1}K^{−1}the specific heat capacity of ice *l*_{fus}= 334 kJkg^{−1}=3.34×10^{5}Jkg^{−1}the specific latent heat of fusion of ice *c*_{W}= 4.18 kJkg^{−1}K^{−1}= 4180 Jkg^{−1}K^{−1}the specific heat capacity of water #### Analysis

First, we will determine how much energy is released when “burning” 5 g of fat. The energy released is equal to the heat received by the ice. This heat is a sum of the heat required to warm the ice to the melting temperature, the latent heat needed to melt the ice and the heat required to warm the melted water to the body temperature. Hence we can determine the mass of ice and consequently the number of cubes.

#### Solution

The energy released by burning fat of a mass

*m*_{f}:\[E=Hm_f.\]The heat that needs to be supplied to ice of unknown mass

*m*_{I}to be melted and warmed up to the body temperature:the warming of ice: \[Q_1 = c_I m_I (t_m - t_I)\]the melting of ice: \[L_I = l_{fus} m_I\]the warming of melted water: \[Q_2 = c_W m_I(t_B - t_m)\]**In total:**: \[Q= c_I m_I (t_m - t_I) +l_{fus} m_L + c_W m_I(t_B - t_m)=\]\[\ \, \ = m_I[c_I (t_m-t_I) + l_{fus} + c_W (t_B - t_m)].\]The energy gained by burning fat is used for warming the ice, that is to say:

\[E=Q,\]\[Hm_{f}=m_I [c_I (t_m-t_I) +l_{fus} + c_W (t_B - t_m)].\]Now we can evaluate the unknown mass of ice and then substitute the numerical values:

\[m_I = \frac{Hm_f}{c_I (t_m-t_I) + l_{fus} + c_W (t_B - t_m)}\]\[m_I = \frac{38 \cdot{10^6} \cdot 5}{2100 \cdot [0-(-20)] + 3.34 \cdot{10^5} + 4180 \cdot (36.8 - 0)}\,\mathrm kg \dot= \]\[\ \, \ \dot= 360 \,\mathrm kg \]Finally, we determine the number of ice cubes:

\[N = \frac{m_I}{m_1}=\frac{Hm_f}{m_1[c_I (t_m-t_I) + l_{fus} + c_W (t_B - t_m)]}\]\[N \dot= 7200\]#### Answer

When on the “ice diet”, you would have to eat about 7200 ice cubes to lose five kilograms of fat, which corresponds to about 360 kilograms of ice.